Lie symmetries of the canonical geodesic equations for six-dimensional nilpotent lie groups

Abstract

For each of the six-dimensional indecomposable nilpotent Lie algebras, the geodesic equations of the associated canonical Lie group connection are given. In each case, a basis for the associated Lie algebra of symmetries is constructed and analyzed.

Keywords:

• Canonical lie group connection
• canonical geodesic equations
• lie symmetry algebra

PUBLIC INTEREST STATEMENT

Lie symmetry methods are used to construct invariant solutions of differential equations. Given a differential equation (or a partial differential equation), the objective would be to find the group of infinitesimal transformations. These transformations are then used to construct invariant solutions for the given equations.

1. Introduction

In this paper, we continue with our investigation of the Lie symmetries of the geodesic system of the natural linear torsion-free connection associated with any Lie group. See Cartan & Schouten (1926), Ghanam et al. (2004) and Helgason (1978) for further details about this connection. In Ghanam et al. (2004) and Helgason (1978), the main geometrical properties of this connection have been listed and proofs supplied. The articles (Almusawa, Ghanam, Thompson et al., 2019a, 2019b; R. Ghanam & Thompson, 0000a, 0000b) were concerned with geodesic systems in dimensions two, three, four and five, respectively. The present article extends the investigation to indecomposable nilpotent groups and Lie algebras in dimension six. By an indecomposable Lie algebra, we mean that one is not a direct sum of lower dimensional Lie algebras. Altogether, in dimension six there are around 163 classes of solvable, indecomposable Lie algebras, depending on how one counts them. To analyze the class of six-dimensional, indecomposable nilpotent Lie algebras presents already quite a substantial challenge and this article is devoted to this class. Matrix representations for all such groups and algebras may be found in, for example, R. Ghanam & Thompson (2018). There are 24 such Lie algebras up to isomorphism, none of which contain parameters, and we use the numbering given in Patera et al. (1976). We use $\mathfrak{g}\mathfrak{h}$ to denote a semi-direct product of Lie algebras in which g is a subalgebra and $\mathfrak{h}$ is an ideal. For each nilpotent Lie algebra in dimension six, we list a matrix Lie group associated with the given Lie algebra, a list of right-invariant vector fields representing the Lie algebra, the geodesic equations and a basis for the symmetry algebra. We then analyze the algebra using its Levi decomposition. Ideally, we would like to give the Lie brackets for all the symmetry algebras concerned, however, to do so would make this article unnecessarily lengthy and the long lists would not offer the reader much insight.

Above and beyond the goal of obtaining the Lie symmetry algebras for each of the 24 nilpotent six-dimensional Lie groups, another goal of this paper is to develop some familiarity with handling Lie groups and Lie algebras in a relatively high dimension. Apart from theoretical investigations that work in arbitrary dimensions, most Lie algebra calculations hitherto have of necessity been confined to small dimensions, mostly less than 10.

2. Flat connection cases

Let $M$ be a smooth manifold endowed with a symmetric connection. If the connection is flat, then its geodesic equations are, in some system of local coordinates on $M$,

(1) ${\ddot{x}}^i=0,$(1)

where $1\le i\le n$ and $n$ is the dimension of $M$. If we define the dilation vector field $\mathrm{\Delta }$ on $M$ by

(2) $\mathrm{\Delta }=t{D}_t+x^i{D}_i,$(2)

where $D_i$ denotes the partial derivative operator with respect to $x^i$ and there is a sum over $i$, then the following vector fields comprise a basis for the space of Lie symmetries of Equation (1)(1) ${\ddot{x}}^i=0,$(1) :

(3) $D_t,D_i,t{D}_t,x^i{D}_t,t{D}_i,x^i{D}_j,t\mathrm{\Delta },x^i\mathrm{\Delta }.$(3)

The dimension of the Lie algebra of the symmetries of a flat connection in dimension $n$ is $n^2+4n+3=(n+2{)}^2-1$. In fact, the Lie algebra of vector fields in Equation (3)(3) $D_t,D_i,t{D}_t,x^i{D}_t,t{D}_i,x^i{D}_j,t\mathrm{\Delta },x^i\mathrm{\Delta }.$(3) is $\mathfrak{s}\mathfrak{l}(n+2,\mathbb{R})$.

In the case of the six-dimensional nilpotent Lie algebras, there are three cases where the connection is flat, namely, 6.3, 6.4 and 6.5 so the symmetry algebra is $\mathfrak{s}\mathfrak{l}(8,\mathbb{R})$. In each case below, we list the non-zero brackets for the algebra and the geodesic equations. It is important to emphasize, however, that the system of coordinates in which the right-hand sides of the geodesic equations are all zero, is not compatible with the Lie algebra structure; changing to such a system of local coordinates does nonetheless demonstrate that the symmetry algebra is $\mathfrak{s}\mathfrak{l}(n+2,\mathbb{R})$.

6.3: $[e_1,e_2]=e_6,[e_1,e_3]=e_4,[e_2,e_3]=e_5$:

$S=\left[\begin{array}{ccccc}1& 0& z& \frac{z^2}{2}& p\\ 0& 1& q& x& y\\ 0& 0& 1& z& w\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$

Right-invariant vector fields: $D_w,D_z+w{D}_p,D_q+w{D}_y+z{D}_x,D_y,D_x,D_p$.

Geodesics:

(4) $\begin{array}{c}\ddot{x}=\dot{z}\dot{q}\\ \ddot{y}=\dot{w}\dot{q}\\ \ddot{z}=0\\ \ddot{w}=0\\ \ddot{p}=\dot{z}\dot{w}\\ \ddot{q}=0.\end{array}$(4)

If we make the following change of variables

$\bar{x}=x-\frac{1}{2}zq,\bar{y}=y-\frac{1}{2}wq,\bar{z}=z,\bar{w}=w,\bar{p}=p-\frac{1}{2}zw,\bar{q}=q,$

then the geodesic equations become:

(5) $\begin{array}{l}\ddot{\overline{x}}=0\\ \ddot{\overline{y}}=0\\ \ddot{\overline{z}}=0\\ \ddot{\overline{w}}=0\\ \ddot{\overline{p}}=0\\ \ddot{\overline{q}}=0.\end{array}$(5)

6.4: $[e_1,e_2]=e_5,[e_1,e_3]=e_6,[e_2,e_4]=e_6$:

$S=\left[\begin{array}{cccccc}1& x& y& q& p& \hspace{0.17em}\\ 0& 1& 0& z& y& \hspace{0.17em}\\ 0& 0& 1& w& 0& \hspace{0.17em}\\ 0& 0& 0& 1& 0& \hspace{0.17em}\\ 0& 0& 0& 0& 1& \hspace{0.17em}\end{array}\right].$

Right-invariant vector fields: $-D_x-y{D}_p-z{D}_q,D_y+w{D}_q,D_z,-D_w,D_p,D_q.$

Geodesics:

(6) $\begin{array}{c}\ddot{x}=0\\ \ddot{y}=0\\ \ddot{z}=0\\ \ddot{w}=0\\ \ddot{p}=\dot{x}\dot{y}\\ \ddot{q}=\dot{x}\dot{z}+\dot{y}\dot{w}.\end{array}$(6)

If we make the following change of variables

$\bar{x}=x\bar{y}=y,\bar{z}=z,\bar{w}=w,\bar{p}=p-\frac{1}{2}xy,\bar{q}=q-\frac{1}{2}(xz+yw),$

then the geodesic equations become:

(7) $\begin{array}{l}\ddot{\overline{x}}=0\\ \ddot{\overline{y}}=0\\ \ddot{\overline{z}}=0\\ \ddot{\overline{w}}=0\\ \ddot{\overline{p}}=0\\ \ddot{\overline{q}}=0.\end{array}$(7)

6.5: $[e_1,e_3]=e_5,[e_1,e_4]=e_6,[e_2,e_3]=-e_6,[e_2,e_4]=e_5$:

$S=\left[\begin{array}{cccccc}1& x& y& q& p& \hspace{0.17em}\\ 0& 1& 0& w& z& \hspace{0.17em}\\ 0& 0& 1& -z& w& \hspace{0.17em}\\ 0& 0& 0& 1& 0& \hspace{0.17em}\\ 0& 0& 0& 0& 1& \hspace{0.17em}\end{array}\right].$

Right-invariant vector fields: $D_w,D_z,D_y-z{D}_q+w{D}_p,D_x+z{D}_p+w{D}_q,D_p,D_q$.

Geodesics:

(8) $\begin{array}{c}\ddot{x}=0\\ \ddot{y}=0\\ \ddot{z}=0\\ \ddot{w}=0\\ \ddot{p}=\dot{x}\dot{z}+\dot{y}\dot{w}\\ \ddot{q}=\dot{x}\dot{w}-\dot{y}\dot{z}.\end{array}$(8)

If we make the following change of variables

$\bar{x}=x,\bar{y}=y,\bar{z}=z,\bar{w}=w,\bar{p}=p-\frac{1}{2}(xz+yw),\bar{q}=q-\frac{1}{2}(xw-yz),$

then the geodesic equations become:

(9) $\begin{array}{l}\ddot{\overline{x}}=0\\ \ddot{\overline{y}}=0\\ \ddot{\overline{z}}=0\\ \ddot{\overline{w}}=0\\ \ddot{\overline{p}}=0\\ \ddot{\overline{q}}=0.\end{array}$(9)

3. Non-flat cases

6.1: $[e_1,e_2]=e_3,[e_1,e_3]=e_4,[e_1,e_5]=e_6:$

$S=\left[\begin{array}{cccccc}1& w& \frac{w^2}{2}& x& p& \hspace{0.17em}\\ 0& 1& w& y& q& \hspace{0.17em}\\ 0& 0& 1& z& 0& \hspace{0.17em}\\ 0& 0& 0& 1& 0& \hspace{0.17em}\\ 0& 0& 0& 0& 1& \hspace{0.17em}\end{array}\right].$

Right-invariant vector fields: $-q{D}_p-y{D}_x-z{D}_y-D_w,D_z,D_y,D_x,D_q,D_p$.

The geodesic equations are:

(10) $\begin{array}{c}\ddot{x}=\dot{y}\dot{w}\\ \ddot{y}=\dot{z}\dot{w}\\ \ddot{z}=0\\ \ddot{w}=0\\ \ddot{p}=\dot{w}\dot{q}\\ \ddot{q}=0.\end{array}$(10)

The symmetry algebra is spanned by

(11) $\begin{array}{cc}{e}_1& =D_z,e_2=D_w,e_3=t{D}_t,e_4=D_t,e_5=t{D}_p,e_6=D_p,e_7=t{D}_x,e_8=D_x,\\ e_9& =D_y,e_{10}=D_q,e_{11}=q{D}_t,e_{12}=w{D}_t,e_{13}=z{D}_t,e_{14}=q{D}_p,\\ e_{15}& =z{D}_p,e_{16}=w{D}_p{e}_{17}=q{D}_x,e_{18}=z{D}_x,\\ e_{19}& =w{D}_x,e_{20}=p{D}_p+q{D}_q,e_{21}=p{D}_x+q{D}_y,\\ e_{22}& =y{D}_x+z{D}_y,e_{23}=x{D}_x+y{D}_y+z{D}_z,\\ e_{24}& =tw{D}_x+2t{D}_y,e_{25}=\frac{tw}{2}{D}_p+t{D}_q,\\ e_{26}& =\frac{w^2}{2}{D}_x+w{D}_y,e_{27}=\frac{w^2}{2}{D}_p+w{D}_q,\\ e_{28}& =p{D}_p+w{D}_w+2x{D}_x+y{D}_y,e_{29}=wq{D}_x+2q{D}_y,\\ e_{30}& =wz{D}_x+2z{D}_y,e_{31}=(qw-2p)D_t,e_{32}=(wz-2y)D_t.\\ e_{33}& =(qw-2p)D_p,e_{34}=\frac{wz}{2}{D}_p+z{D}_q,\\ e_{35}& =(-\frac{wz}{2}+y)D_p,e_{36}=\frac{w^3}{6}{D}_x+\frac{w^2}{2}{D}_y+w{D}_z,\\ e_{37}& =(\frac{w^{2}q}{2}-pw)D_p+(qw-2p)D_q,e_{38}=(pw-\frac{q{w}^2}{2})D_x+(-qw+2p)D_y,\\ e_{39}& =(yw-\frac{w^{2}z}{2})D_x+(2y-wz)D_y,e_{40}=(\frac{w^{2}z}{2}-wy)D_p+(wz-2y)D_q.\end{array}$(11)

The symmetry algebra g is a 40-dimensional indecomposable Lie algebra. The Levi decomposition $\mathrm{g}={\mathrm{g}}_1\times {\mathrm{g}}_2$, where ${\mathrm{g}}_1$ is $\mathrm{s}(4,\mathrm{R})$ and spanned by

(12) $\begin{array}{rl}& e_3-\frac{e_{39}}{2},e_5,e_{11},e_{14},e_{20}-e_{39},e_{24},e_{25},e_{29},e_{31},e_{32},\\ & e_{33}+e_{39},e_{35},e_{37},e_{38},e_{40}.\end{array}$(12)

The solvable part ${\mathrm{g}}_2$ is a semidirect product of a 22-dimensional nilradical ${\mathrm{n}}_{22}$ spanned by

(13) $\begin{array}{rl}& e_1,e_2+\frac{e_{14}}{2},e_4,e_6,e_7,e_8,e_9,e_{10},e_{12},\\ & e_{13}{e}_{15},e_{16},e_{17},e_{18},e_{19},e_{21}-\frac{e_{29}}{2},e_{22},\\ & e_{26},e_{27},e_{30},e_{34},e_{36}\end{array}$(13)

and $\mathbb{R}{}^3$ spanned by

(14) $\begin{array}{cc}{e}_3+e_{20}+\frac{e_{39}}{2}& ,e_{23}-\frac{e_{39}}{2},e_{28}+\frac{e_{33}}{2}-\frac{e_{39}}{2}.\end{array}$(14)

We conclude that the symmetry Lie algebra is$\mathrm{s}(4,\mathrm{R})\times ({\mathrm{n}}_{22}\times {\mathrm{R}}^3)$.

6.2:$[e_1,e_2]=e_3,[e_1,e_3]=e_4,[e_1,e_4]=e_5,[e_1,e_5]=e_6$:

$S=\left[\begin{array}{cccccc}1& w& \frac{w^2}{2}& \frac{w^3}{6}& \frac{w^4}{24}& p\\ 0& 1& w& \frac{w^2}{2}& \frac{w^3}{6}& q\\ 0& 0& 1& w& \frac{w^2}{2}& x\\ 0& 0& 0& 1& w& y\\ 0& 0& 0& 0& 1& z\\ 0& 0& 0& 0& 0& 1\end{array}\right].$

Right-invariant vector fields: $-q{D}_p-x{D}_q-y{D}_x-z{D}_y-D_w,D_z,D_y,D_x,D_q,D_p$.

The geodesic equations are:

(15) $\begin{array}{c}\ddot{x}=\dot{y}\dot{w}\\ \ddot{y}=\dot{z}\dot{w}\\ \ddot{z}=0\\ \ddot{w}=0\\ \ddot{p}=\dot{w}\dot{q}\\ \ddot{q}=\dot{x}\dot{w}.\end{array}$(15)

The symmetry algebra is spanned by

(16) $\begin{array}{cc}{e}_1& =D_t,e_2=t{D}_p,e_3=D_p,e_4=D_x,e_5=D_y,e_6=D_q,e_7=D_z,e_8=D_w,\\ e_9& =t{D}_t,e_{10}=z{D}_p,e_{11}=w{D}_p,e_{12}=w{D}_t,e_{13}=z{D}_t,e_{14}=y{D}_p+z{D}_q,\\ e_{15}& =x{D}_p+y{D}_q+z{D}_x,e_{16}=tw{D}_p+2t{D}_q,e_{17}=q{D}_p+x{D}_q+y{D}_x+z{D}_y,\\ e_{18}& =\frac{w^2}{2}{D}_p+w{D}_q,e_{19}=p{D}_p+q{D}_q+x{D}_x+y{D}_y+z{D}_z,e_{20}=wz{D}_p+2D_q,e_{21}=(wz-2y)D_t,\\ e_{22}& =4p{D}_p+3q{D}_q+w{D}_w+2x{D}_x+y{D}_y,e_{23}=\frac{w^3}{6}{D}_p+\frac{w^2}{2}{D}_q+w{D}_x,\\ e_{24}& =\frac{w^4}{24}{D}_p+\frac{w^3}{6}{D}_q+\frac{w^2}{2}x+w{D}_y,e_{25}=(-\frac{w^{2}z}{2}+yw)D_p+(2y-wz)D_q,\\ e_{26}& =\frac{w^5}{120}{D}_p+\frac{w^4}{24}{D}_q+\frac{w^3}{6}{D}_x+\frac{w^2}{2}{D}_y+w{D}_z,\\ e_{27}& =1/6w(w^{2}z-3wy+6x)D_p+(2x-wy+1/2z{w}^2)D_q+wz{D}_x+2z{D}_y\\ e_{28}& =(-\frac{z{w}^4}{24}+\frac{y{w}^3}{6}-\frac{x{w}^2}{2}+qw)D_p+(2q-wx+\frac{w^{2}y}{2}-\frac{w^{3}z}{6})D_q+(-\frac{z{w}^2}{2}+wy)D_x+(-wz+2y)D_y.\end{array}$(16)

The symmetry algebra g is a 28-dimensional indecomposable solvable Lie algebra. It has a 24-dimensional nilradical ${\mathrm{n}}_{24}$ spanned by

(17) $\begin{array}{cc}{e}_1& ,e_2,e_3,e_4,e_5,e_6,e_7,e_8,e_{10},e_{11},e_{12},e_{13},e_{14},\\ e_{15}& ,e_{16},e_{17},e_{18},e_{20},e_{21},e_{23},e_{24},e_{25},e_{26},e_{27},\end{array}$(17)

and a four-dimensional abelian complement $R{}^4$ spanned by

(18) $\begin{array}{cc}{e}_9& ,e_{19},e_{22},e_{28},\end{array}$(18)

We conclude that the symmetry Lie algebra is${\mathrm{n}}_{24}\times {\mathrm{R}}^4$.

6.6: $[e_1,e_2]=e_6,[e_1,e_3]=e_4,[e_1,e_4]=e_5,[e_2,e_3]=e_5$:

$S=\left[\begin{array}{ccccc}1& q& w& x& p\\ 0& 1& q& y& \frac{q^2}{2}\\ 0& 0& 1& z& q\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$

Right-invariant vector fields: $D_q+z{D}_y+y{D}_x+q{D}_w+\frac{q^2}{2}{D}_p,-(D_w+z{D}_x+q{D}_p),D_z,-D_y,D_x,-D_p$.

The geodesic equations are:

(19) $\begin{array}{c}\ddot{x}=-q\dot{q}\dot{z}+\dot{q}\dot{y}+\dot{z}\dot{w}\\ \ddot{y}=\dot{q}\dot{z}\\ \ddot{z}=0\\ \ddot{w}={\dot{q}}^2\\ \ddot{p}=\dot{w}\dot{q}\\ \ddot{q}=0\end{array}$(19)

The symmetry algebra is spanned by

(20) $\begin{array}{cc}{e}_1& =D_t,e_2=D_w,e_3=t{D}_p,e_4=D_p,e_5=D_z,e_6=D_y,e_7=t{D}_x,e_8=D_x,\\ e_9& =t{D}_t,e_{10}=z{D}_t,e_{11}=q{D}_t,e_{12}=z{D}_p,e_{13}=q{D}_p,e_{14}=z{D}_x,e_{15}=q{D}_x,\\ e_{16}& =\frac{tq}{2}{D}_x+t{D}_y,e_{17}=\frac{q^2}{2}{D}_x+q{D}_y,e_{18}=(-\frac{q^2}{2}+w)D_p,e_{19}=(-\frac{q^2}{2}+w)D_x,\\ e_{20}& =(-\frac{q^2}{2}+w)D_t,e_{21}=D_q-\frac{qz}{2}{D}_x,e_{22}=p{D}_x+\frac{q^2}{2}{D}_y+q{D}_z,e_{23}=\frac{qz}{2}{D}_x+z{D}_y,\\ e_{24}& =(-\frac{qz}{2}+y)D_p,e_{25}=(-\frac{qz}{2}+y)D_x,e_{26}=(-\frac{qz}{2}+y)D_t,e_{27}=\frac{tq}{2}{D}_p+t{D}_w+\frac{tz}{2}{D}_x,\\ e_{28}& =(-\frac{q^3}{6}+qw-2p)D_p,e_{29}=(-\frac{q^3}{6}+qw-2p)D_x,e_{30}=(-\frac{q^3}{6}+qw-2p)D_t,\\ e_{31}& =\frac{qz}{2}{D}_p+z{D}_w+\frac{z^2}{2}{D}_x,e_{32}=\frac{q^2}{2}{D}_p+q{D}_w+\frac{qz}{2}{D}_x,e_{33}=(-\frac{q^3}{6}+p)D_x+(-\frac{q^2}{2}+w)D_y,\\ e_{34}& =-\frac{q(qz-2y)}{4}{D}_x+(-\frac{qz}{2}+y)D_y,e_{35}=(\frac{z{q}^2}{4}-\frac{qy}{2}+x)D_x+\frac{qz}{2}{D}_y+z{D}_z,\\ e_{36}& =(-\frac{q^3}{6}+p)D_p+(-\frac{q^2}{2}+w)D_w-\frac{z(q^2-2w)}{4}{D}_x,e_{37}=(-\frac{q^4}{12}+\frac{w{q}^2}{2}-pq)D_x+(-\frac{q^3}{6}+qw-2p)D_y,\\ e_{38}& =(-\frac{q(qz-2y)}{4})D_p+(-\frac{qz}{2}+y)D_w-\frac{z(qz-2y)}{4}{D}_x,\\ e_{39}& =(p+\frac{q^3}{3})D_p+q{D}_q+q^2{D}_w+((\frac{3}{4}{q}^2-w)z-\frac{qy}{2}+2x)D_x+\frac{qz}{2}{D}_y,\\ e_{40}& =(-\frac{q^4}{12}+\frac{q^{2}w}{2}-pq)D_p+(-\frac{q^3}{6}+qw-2p)D_w-\frac{1}{12}z(q^3-6qw+12p)D_x.\end{array}$(20)

The symmetry algebra g is a 40-dimensional indecomposable Lie algebra. The Levi decomposition $\mathrm{g}={\mathrm{g}}_1\times {\mathrm{g}}_2$, where ${\mathrm{g}}_1$ is $\mathrm{s}(4,\mathrm{R})$ and spanned by

(21) $\begin{array}{rl}& e_3,e_9-\frac{1}{2}{e}_{36},e_{16},e_{18},e_{20},e_{24},e_{26},e_{27},e_{28}+e_{36},e_{29}+2e_{33},\\ & e_{30},e_{34}-\frac{1}{2}{e}_{36},e_{37},e_{38},e_{40}.\end{array}$(21)

The solvable part ${\mathrm{g}}_2$ is a semidirect product of a 22-dimensional nilradical ${\mathrm{n}}_{22}$ spanned by

(22) $\begin{array}{cc}{e}_1,& e_2,e_4,e_5,e_6,e_7,e_8,e_{10},e_{11},\\ e_{12}& e_{13},e_{14},e_{15},e_{17},e_{18}+2e_{21},e_{19},e_{22},\\ e_{23},& e_{25},e_{29},e_{31},e_{32}\end{array}$(22)

and $\mathbb{R}{}^3$ spanned by

(23) $\begin{array}{cc}{e}_9+e_{34}+e_{36}& ,e_{28}+2e_{39},e_{35}.\end{array}$(23)

We conclude that the symmetry Lie algebra is$\mathrm{s}(4,\mathrm{R})\times ({\mathrm{n}}_{22}\times {\mathrm{R}}^3)$.

6.7: $[e_1,e_3]=e_4,[e_1,e_4]=e_5,[e_2,e_3]=e_6$:

$S=\left[\begin{array}{ccccc}1& 0& 0& p& q\\ 0& 1& z& \frac{z^2}{2}& x\\ 0& 0& 1& z& y\\ 0& 0& 0& 1& w\\ 0& 0& 0& 0& 1\end{array}\right].$

Right-invariant vector fields: $-(D_w+z{D}_y+y{D}_x),-(D_q+z{D}_p),D_z,D_y,D_x,D_p$.

The geodesic equations are:

(24) $\begin{array}{c}\ddot{x}=\dot{y}\dot{w}\\ \ddot{y}=\dot{z}\dot{w}\\ \ddot{z}=0\\ \ddot{w}=0\\ \ddot{p}=\dot{z}\dot{q}\\ \ddot{q}=0\end{array}$(24)

The symmetry algebra is spanned by

(25) $\begin{array}{cc}{e}_1& =D_z,e_2=t{D}_t,e_3=D_t,e_4=t{D}_p,e_5=D_p,e_6=t{D}_x,e_7=t{D}_x,e_8=D_y,\\ e_9& =D_q,e_{10}=D_w,e_{11}=q{D}_t,e_{12}=w{D}_t,e_{13}=z{D}_t,e_{14}=q{D}_p,e_{15}=w{D}_p,\\ e_{16}& =z{D}_p,e_{17}=q{D}_x,e_{18}=z{D}_x,e_{19}=w{D}_x,\\ e_{20}& =p{D}_p+q{D}_q,e_{21}=y{D}_x+z{D}_y,e_{22}=y{D}_p+w{D}_q,e_{23}=wt{D}_x+2t{D}_y,\\ e_{24}& =w{D}_w+2x{D}_x+y{D}_y,e_{25}=p{D}_p+x{D}_x+y{D}_y+z{D}_z,e_{26}=\frac{tz}{2}{D}_p+t{D}_q,e_{27}=\frac{w^2}{2}{D}_x+w{D}_y,\\ e_{28}& =\frac{z^2}{2}{D}_p+z{D}_q,e_{29}=wq{D}_x+2q{D}_y,e_{30}=zw{D}_x+2z{D}_y,\\ e_{31}& =(qz-2p)D_t,e_{32}=(wz-2y)D_x,e_{33}=(qz-2p)D_p,\\ e_{34}& =(wz-2y)D_p,e_{35}=(qz-2p)D_x,\\ e_{36}& =\frac{qw}{2}{D}_p+\frac{w^3}{6}{D}_x+\frac{w^2}{2}{D}_y+w{D}_z,e_{37}=(\frac{q{z}^2}{2}-pz)D_p+(qz-2p)D_q,\\ e_{38}& =(-\frac{w^{2}z}{2}+wy)D_x+(-wz+2y)D_y,\\ e_{39}& =(\frac{w{z}^2}{2}-yz)D_p+(wz-2yq)D_q,\\ e_{40}& =(qzw-2pw)D_x+(2qz-4p)D_y.\end{array}$(25)

The symmetry algebra g is a 40-dimensional indecomposable Lie algebra. The Levi decomposition $g=g_1\times {\mathrm{g}}_2$, where $g_1$ is $\mathrm{s}(4,\mathrm{R})$ and spanned by

(26) $\begin{array}{rl}& e_2-\frac{1}{2}{e}_{38},e_4,e_{11},e_{14},e_{20}-e_{38},e_{23},e_{26},e_{29},e_{31},e_{32},\\ & e_{33}+e_{38},e_{34},e_{37},e_{39},e_{40}.\end{array}$(26)

The solvable part ${\mathrm{g}}_2$ is a semidirect product of a 22-dimensional nilradical ${\mathrm{n}}_{22}$ spanned by

(27) $\begin{array}{cc}{e}_1+\frac{1}{2}{e}_{14},& e_3,e_5,e_6,e_7,e_8,e_9,e_{10},e_{12},\\ e_{13}& e_{15},e_{16},e_{17},e_{18},e_{19},e_{21},e_{22}+\frac{1}{2}{e}_{34},\\ e_{27},& e_{28},e_{30},e_{35},e_{36}\end{array}$(27)

and $\mathbb{R}{}^3$ spanned by

(28) $\begin{array}{cc}{e}_2+e_{20}+\frac{1}{2}{e}_{38}& ,e_{24}-\frac{1}{2}{e}_{38},e_{25}+\frac{1}{2}{e}_{38}.\end{array}$(28)

We conclude that the symmetry Lie algebra is$\mathrm{s}(4,\mathrm{R})\times ({\mathrm{n}}_{22}\times {\mathrm{R}}^3)$.

6.8: $[e_1,e_2]=e_3+e_5,[e_1,e_3]=e_4,[e_2,e_5]=e_6$:

$S=\left[\begin{array}{cccccc}1& 0& -\frac{1}{2}z& p+\frac{1}{2}y-\frac{1}{2}zw& q& \hspace{0.17em}\\ 0& 1& w& \frac{1}{2}{w}^2& x& \hspace{0.17em}\\ 0& 0& 1& w& y& \hspace{0.17em}\\ 0& 0& 0& 1& z& \hspace{0.17em}\\ 0& 0& 0& 0& 1& \hspace{0.17em}\end{array}\right].$

Right-invariant vector fields: $D_z-\frac{y}{2}{D}_q,-(z{D}_y+y{D}_x+D_w),-D_p-z{D}_q,-D_q,\frac{z}{2}{D}_q-D_y+D_p,-D_x$.

The geodesic equations are:

(29) $\begin{array}{c}\ddot{x}=\dot{y}\dot{w}\\ \ddot{y}=\dot{z}\dot{w}\\ \ddot{z}=0\\ \ddot{w}=0\\ \ddot{p}=0\\ \ddot{q}=\dot{z}\dot{p}-\frac{1}{2}\dot{z}\dot{w}\end{array}$(29)

The symmetry algebra is spanned by

(30) $\begin{array}{cc}{e}_1& =D_z,e_2=w{D}_w,e_3=t{D}_t,e_4=D_t,e_5=t{D}_x,e_6=D_x,e_7=D_y,e_8=t{D}_q,\\ e_9& =D_q,e_{10}=D_p,e_{11}=p{D}_t,e_{12}=w{D}_t,e_{13}=z{D}_t,e_{14}=p{D}_x,e_{15}=z{D}_x,\\ e_{16}& =w{D}_x,e_{17}=p{D}_q,e_{18}=w{D}_q,e_{19}=z{D}_q,\\ e_{20}& =y{D}_x+z{D}_y,e_{21}=tw{D}_x+2t{D}_y,e_{22}=q{D}_q+x{D}_x+y{D}_y+z{D}_z,e_{23}=t{D}_p+\frac{tz}{2}{D}_q,\\ e_{24}& =\frac{w^2}{2}{D}_x+w{D}_y,e_{25}=z{D}_p+\frac{z^2}{2}{D}_q,e_{26}=pw{D}_x+2p{D}_y,e_{27}=wz{D}_x+2z{D}_y,\\ e_{28}& =(wz-2y)D_t,e_{29}=w{D}_p+\frac{wz}{2}{D}_q,e_{30}=(-\frac{wz}{2}+y)D_q,\\ e_{31}& =(pz-2q)D_x+z{D}_y,e_{32}=(pz-2q-y)D_t,e_{33}=p{D}_p+(\frac{wz}{4}+q)D_q,\\ e_{34}& =-\frac{wz}{4}{D}_q+w{D}_w+2x{D}_x+y{D}_y,e_{35}=((p-\frac{w}{2})z-2q)D_q,\\ e_{36}& =(\frac{w^{2}z}{2}+wy)D_x+(2y-wz)D_y,e_{37}=(wz-2y)D_p+(\frac{w{z}^2}{2}-yz)D_q,\\ e_{38}& =1/4w(2p-w)D_q+1/6w^3{D}_x+1/2w^2{D}_y+w{D}_z,\\ e_{39}& =(pz-2q-y)D_p+1/2z(pz-2q-y)D_q,\\ e_{40}& =w((p-1/2w)z-2q)D_x+((2p-w)z-4q)D_y.\end{array}$(30)

The symmetry algebra g is a 40-dimensional indecomposable Lie algebra. The Levi decomposition $g=g_1\times {\mathrm{g}}_2$, where $g_1$ is $\mathrm{s}(4,\mathrm{R})$ and spanned by

(31) $\begin{array}{rl}& e_3-\frac{1}{2}{e}_{36},e_8,e_{11},e_{17},e_{21},e_{23},e_{26},e_{28},e_{30},e_{32},\\ & e_{33}-e_{36},e_{35}+e_{36},e_{37},e_{39},e_{40}.\end{array}$(31)

The solvable part ${\mathrm{g}}_2$ is a semidirect product of a 22-dimensional nilradical ${\mathrm{n}}_{22}$ spanned by

(32) $\begin{array}{cc}{e}_1+\frac{1}{2}{e}_{17},& e_2,e_4,e_5,e_6,e_7,e_9,e_{10},e_{12},\\ e_{13}& e_{14},e_{15},e_{16},e_{18},e_{19},e_{20},e_{24},\\ e_{25},& e_{27},e_{29},e_{31},e_{38}\end{array}$(32)

and $\mathbb{R}{}^3$ spanned by

(33) $\begin{array}{cc}{e}_3+e_{33}+\frac{1}{2}{e}_{36}& ,e_{22}+\frac{1}{2}{e}_{35}-\frac{1}{2}{e}_{36},e_{34}-\frac{1}{2}{e}_{36}.\end{array}$(33)

We conclude that the symmetry Lie algebra is$\mathrm{s}(4,\mathrm{R})\times ({\mathrm{n}}_{22}\times {\mathrm{R}}^3)$.

6.9: $[e_1,e_2]=e_3,[e_1,e_3]=e_4,[e_1,e_5]=e_6,[e_2,e_3]=e_6$:

$S=\left[\begin{array}{ccccc}1& 0& w-y& z& p\\ 0& 1& 2w-y& wy-x+2z& q\\ 0& 0& 1& w& x\\ 0& 0& 0& 1& y\\ 0& 0& 0& 0& 1\end{array}\right].$

Right-invariant vector fields: $D_w+y{D}_x+w{D}_z+x{D}_p+2x{D}_q,-D_x+y{D}_q,D_y-w{D}_z-x{D}_p-x{D}_q,D_p,-D_z-y{D}_p-2y{D}_q,D_q$.

The geodesic equations are:

(34) $\begin{array}{c}\ddot{x}=\dot{y}\dot{w}\\ \ddot{y}=0\\ \ddot{z}=-\dot{y}\dot{w}\\ \ddot{w}=0\\ \ddot{p}=w{\dot{y}}^2-y{\dot{w}}^2+\dot{x}\dot{w}-\dot{x}\dot{y}+\dot{y}\dot{z}-w\dot{y}\dot{w}\\ \ddot{q}=(3y-2w))\dot{y}\dot{w}-2y{\dot{w}}^2+2w{\dot{y}}^2+2\dot{y}\dot{z}+2\dot{x}\dot{w}.\end{array}$(34)

The symmetry algebra is spanned by

(35) $\begin{array}{cc}{e}_1& =t{D}_t,e_2=D_z,e_3=D_x,e_4=t{D}_p,e_5=D_p,e_6=t{D}_q,e_7=D_q,e_8=D_t,\\ e_9& =w{D}_p,e_{10}=y{D}_p,e_{11}=w{D}_t,e_{12}=y{D}_q,e_{13}=w{D}_q,e_{14}=y{D}_t,e_{15}=(x+z)D_t,\\ e_{16}& =(yw-2x)D_t,e_{17}=1/2y^2{D}_p+y^2{D}_q+y{D}_z,e_{18}=(x-1/2yw)D_p,e_{19}=(z+1/2yw)D_p,\\ e_{20}& =(x-1/2yw)D_q,e_{21}=(z+1/2yw)D_q,e_{22}=1/2ty{D}_p+ty{D}_q+t{D}_z,\\ e_{23}& =1/2yw{D}_p+D_{q}yw+w{D}_z,\\ e_{24}& =1/2t(-y+w)D_p+tw{D}_q+t{D}_x,e_{25}=1/2w(-y+w)D_p+w^2{D}_q+w{D}_x,\\ e_{26}& =-1/2w^2{D}_p+(-w^2+3/2yw)D_q+D_y,e_{27}=1/2y(-y+w)D_p+yw{D}_q+y{D}_x,\\ e_{28}& =-1/2y(-y+w)D_p-y(-y+w)D_q+D_w,\\ e_{29}& =-1/4y(wy-2x)D_p+(-1/2y^{2}w+xy)D_q+(x-1/2yw)D_z,\\ e_{30}& =1/4y(wy+2z)D_p+(1/2y^{2}w+zy)D_q+(z+1/2yw)D_z,\\ e_{31}& =(1/4y-1/4w)(wy-2x)D_p+(-1/2y{w}^2+xw)D_q+(x-1/2yw)D_x,\\ e_{32}& =(-1/4y+1/4w)(wy+2z)D_p+(1/2y{w}^2+wz)D_q+(z+1/2yw)D_x,\\ e_{33}& =(-1/6y^3-1/2y^{2}w+3/4xy+1/2p-1/4q)D_p+(-y^{2}w+1/2y^3\\ \hspace{0.17em}& +3/2xy+p-1/2q)D_q+y{D}_w+1/2y^2{D}_x-1/2y^2{D}_z,\\ e_{34}& =((1/4w^2+1/2x)y-1/2xw+2p-1/2q)D_p+(1/2y{w}^2+\\ \hspace{0.17em}& (1/2y^2-x)w+q)D_q+1/2yw{D}_x+y{D}_y-1/2yw{D}_z,\\ e_{35}& =(-1/6w^3+y{w}^2+(-y^2-3/2x+1/2z)w-zy+q)D_p\\ \hspace{0.17em}& -1/3w(w^2-9/2yw-3z)D_q+1/2w^2{D}_x+w{D}_y-1/2w^2{D}_z,\\ e_{36}& =(-y^{2}w+(1/4w^2+1/2x-z)y-1/2xw+p+1/2q)D_p\\ \hspace{0.17em}& +(1/2y{w}^2+(-3/2y^2-x)w-2zy+2q)D_q+w{D}_w+1/2yw{D}_x-1/2yw{D}_z.\end{array}$(35)

The symmetry algebra g is a 36-dimensional indecomposable Lie algebra. The Levi decomposition $g=g_1\times {\mathrm{g}}_2$, where $g_1$ is $\mathrm{s}(3,\mathrm{R})\oplus \mathrm{s}(2,\mathrm{R})$ and spanned by

(36) $\begin{array}{cc}{e}_1-e_{31},& e_{15},e_{16},e_{22},e_{24},e_{29},e_{30}-e_{31},e_{32},e_{33},\\ e_{34}-e_{36},& e_{35}.\end{array}$(36)

The solvable part ${\mathrm{g}}_2$ is a semidirect product of a 23-dimensional nilradical ${\mathrm{n}}_{23}$ spanned by

(37) $\begin{array}{cc}{e}_2,& e_3,e_4,e_5,e_6,e_7,e_8,e_9,\\ e_{10}& e_{11},e_{12},e_{13},e_{14},e_{17},e_{18},e_{19},\\ e_{20},& e_{21},e_{23},e_{25},e_{26},e_{27},e_{28}\end{array}$(37)

and $\mathbb{R}{}^2$ spanned by

(38) $\begin{array}{cc}{e}_1+e_{30}+e_{31}& ,e_{34}+e_{36}.\end{array}$(38)

We conclude that the symmetry Lie algebra is$\mathrm{s}(3,\mathrm{R})\oplus \mathrm{s}(2,\mathrm{R})\times ({\mathrm{n}}_{23}\times {\mathrm{R}}^2)$.

6.10: $[e_1,e_2]=e_3,[e_1,e_3]=e_5,[e_1,e_4]=e_6,[e_2,e_3]=a{e}_6,[e_2,e_4]=e_5$: As in case 6.5 $a$ can be reduced to $a=\pm 1$.

$a=1$: The algebra is equivalent to algebra 6.8. The change of basis that changes 6.8 to 6.10($a=1$) is given by:

$\overline{e_1}=e_1+e_2,\overline{e_2}=e_1-e_2,\overline{e_3}=-2(e_3+e_5),\overline{e_4}=2(e_5-e_3),\overline{e_5}=-2(e_4+e_6),$ $\overline{e_6}=2(e_6-e_4).$

$a=-1:$

$S=\left[\begin{array}{cccccc}1& 0& -z& w& y-zw& q\\ 0& 1& w& z& \frac{w^2}{2}& x\\ 0& 0& 1& 0& w& 2y\\ 0& 0& 0& 1& 0& p\\ 0& 0& 0& 0& 1& 2z\\ 0& 0& 0& 0& 0& 1\end{array}\right].$

Right-invariant vector fields: $\sqrt{2}(p{D}_q+2y{D}_x+z{D}_y+D_w,-2y{D}_q+p{D}_x+D_z,\sqrt{2}(-2z{D}_q-D_y),-4D_p,4D_x,4\sqrt{2}{D}_q$.

The geodesic equations are:

(39) $\begin{array}{c}\ddot{x}=\dot{p}\dot{z}+2\dot{w}\dot{y}\\ \ddot{y}=\dot{z}\dot{w}\\ \ddot{z}=0\\ \ddot{w}=0\\ \ddot{p}=0\\ \ddot{q}=6z\dot{z}\dot{w}+\dot{p}\dot{w}.\end{array}$(39)

The symmetry algebra is spanned by

(40) $\begin{array}{cc}{e}_1& =D_w,e_2=t{D}_t,e_3=D_t,e_4=D_y,e_5=t{D}_q,e_6=D_q,e_7=t{D}_x,e_8=D_x,\\ e_9& =D_p,e_{10}=p{D}_t,e_{11}=w{D}_t,e_{12}=z{D}_t,e_{13}=p{D}_q,e_{14}=z{D}_q,e_{15}=w{D}_q,\\ e_{16}& =p{D}_x,e_{17}=z{D}_x,e_{18}=w{D}_x,e_{19}=tw{D}_x+t{D}_y,\\ e_{20}& =w^2{D}_x+w{D}_y,e_{21}=pw{D}_x+p{D}_y,e_{22}=wz{D}_x+z{D}_y,e_{23}=3wz{D}_q+D_z,\\ e_{24}& =(wz-2y)D_t,e_{25}=(y-\frac{wz}{2})D_q,e_{26}=(y-\frac{wz}{2})D_x,e_{27}=t{D}_p+\frac{tw}{2}{D}_q+\frac{tz}{2}{D}_x,\\ e_{28}& =w{D}_p+1/2w^2{D}_q+1/2wz{D}_x,e_{29}=z{D}_p+1/2wz{D}_q+1/2z^2{D}_x,\\ e_{30}& =p{D}_p+1/2pw{D}_q+1/2pz{D}_x,\\ e_{31}& =(-1/2z{w}^2+wy)D_x+(y-1/2wz)D_y,\\ e_{32}& =q{D}_q+w{D}_w+((1/2w^2-p)z-wy+2x)D_x+1/2wz{D}_y,\\ e_{33}& =(-pw+2q)D_q+(1/2z{w}^2-wy+x)D_x+1/2wz{D}_y+z{D}_z\\ e_{34}& =(1/2pz+z^3)D_q+z{D}_w+(-1/12pw+yz+1/6q)D_x+1/2z^2{D}_y\\ e_{35}& =(wz-2y)D_p+(-wy+1/2z{w}^2)D_q+(1/2w{z}^2-yz)D_x\\ e_{36}& =((3w^2-3p)z-6wy+6x)D_q+(1/3w^3+1/2pw)D_x+1/2w^2{D}_y+w{D}_z.\end{array}$(40)

The symmetry algebra g is a 36-dimensional indecomposable Lie algebra. The Levi decomposition is given by $g=g_1\times {\mathrm{g}}_2$, where $g_1$ is $\mathrm{s}(3,\mathrm{R})\oplus \mathrm{s}(2,\mathrm{R})$ and spanned by

(41) $\begin{array}{cc}{e}_2-e_{31}& ,e_{10},e_{19},e_{21},e_{24},e_{27},e_{30}-e_{31},e_{35},\\ e_{32}-e_{33}& ,e_{35},e_{36}.\end{array}$(41)

The solvable part ${\mathrm{g}}_2$ is a semidirect product of a 23-dimensional nilradical ${\mathrm{n}}_{23}$ spanned by

(42) $\begin{array}{cc}{e}_1& ,e_3,e_4,e_5,e_6,e_7,e_8,e_9,\\ e_{11}& ,e_{12},e_{13},e_{14},e_{15},e_{16},e_{17},\\ e_{18}& ,e_{20},e_{22},e_{23},e_{25},e_{26},e_{28},e_{29}\end{array}$(42)

and $\mathbb{R}{}^2$ spanned by

(43) $\begin{array}{cc}{e}_2+e_{30}+e_{31}& ,e_{32}+e_{33}.\end{array}$(43)

We conclude that the symmetry Lie algebra is$\mathrm{s}(3,\mathrm{R})\oplus \mathrm{s}(2,\mathrm{R})\times ({\mathrm{n}}_{23}\times {\mathrm{R}}^2)$.

6.11: $[e_1,e_2]=e_3,[e_1,e_3]=e_4,[e_1,e_4]=e_5,[e_2,e_3]=e_6$:

$S=\left[\begin{array}{ccccc}1& y& x-yz+\frac{y^2}{2}& w& p\\ 0& 1& y-z& yz-\frac{z^2}{2}-x& q\\ 0& 0& 1& z& x\\ 0& 0& 0& 1& y\\ 0& 0& 0& 0& 1\end{array}\right].$

Right-invariant vector fields: $\frac{1}{{2.3}^{\frac{1}{3}}}(2D_y+(2yz-z^2-2x)D_w+2q{D}_p+2x{D}_q),3^{\frac{2}{3}}(D_z+y{D}_x-x{D}_q),3^{\frac{1}{3}}(D_x+z{D}_w+x{D}_p-y{D}_q),$ $D_w+y{D}_p-2D_q,3^{\frac{2}{3}}{D}_p,3(D_w+y{D}_p+D_q)$.

The geodesic equations are:

(44) $\begin{array}{c}\ddot{x}=\dot{y}\dot{z}\\ \ddot{y}=0\\ \ddot{z}=0\\ \ddot{w}=0\\ \ddot{p}={\dot{x}}^2+(x+\frac{1}{2}{z}^2-2yz){\dot{y}}^2+y^2{\dot{z}}^2+(y-z)\dot{x}\dot{y}-2y\dot{x}\dot{z}+(2yz-y^2)\dot{y}\dot{z}+\dot{y}\dot{q}\\ \ddot{q}=-2\dot{x}\dot{y}+\dot{x}\dot{z}+2z{\dot{y}}^2-2y{\dot{z}}^2+(3y-2z)\dot{y}\dot{z}\end{array}$(44)

The symmetry algebra is spanned by

(45) $\begin{array}{cc}{e}_1& =t{D}_w,e_2=D_t,e_3=D_q,e_4=t{D}_p,e_5=D_p,e_6=D_w,e_7=t{D}_t,e_8=y{D}_t,\\ e_9& =z{D}_t,e_{10}=w{D}_w,e_{11}=w{D}_p,e_{12}=z{D}_p,e_{13}=y{D}_p,e_{14}=y{D}_w,e_{15}=z{D}_w,\\ e_{16}& =w{D}_t,e_{17}=x{D}_p+z{D}_q,e_{18}=-x{D}_q+y{D}_x+D_z,e_{19}=1/2y^2{D}_p+D_x,\\ e_{20}& =1/2ty{D}_p+t{D}_q,e_{21}=1/2y^2{D}_p+y{D}_q,e_{22}=(yz-2x)D_p,e_{23}=(yz-2x)D_w,\\ e_{24}& =(yz-2x)D_t,e_{25}=1/2wy{D}_p+w{D}_q,e_{26}=2p{D}_p+(-y^{2}z+xy+2q)D_q+x{D}_x+z{D}_z,\\ e_{27}& =2p{D}_p+(y^{2}z-xy+q)D_q+x{D}_x+y{D}_y,e_{28}=(y^{2}z-y{z}^2-2q)D_p+(-z^2+3x)D_q+D_y,\\ e_{29}& =(1/2y^{2}z-xy)D_p+(yz-2x)D_q,\\ e_{30}& =(-3/2y^{2}z+z^{2}y+xy+2q)D_p+(-2x+1/2z^2)D_q+z{D}_x.\end{array}$(45)

The symmetry algebra g is a 30-dimensional indecomposable Lie algebra. The Levi decomposition is given by$g=g_1\times {\mathrm{g}}_2$, where $g_1$ is $\mathrm{s}(2,\mathrm{R})$ and spanned by

(46) $\begin{array}{cc}{e}_1,& e_7-e_{10},e_{16}.\end{array}$(46)

The solvable part ${\mathrm{g}}_2$ is a semidirect product of a 24-dimensional nilradical ${\mathrm{n}}_{24}$ spanned by

(47) $\begin{array}{cc}{e}_2,& e_3,e_4,e_5,e_6,e_8,e_9,e_{11},e_{12},e_{13},e_{14},e_{15},\\ e_{17},& e_{18},e_{19},e_{20},e_{21},e_{22},e_{23},e_{24},e_{25},e_{28},e_{29},e_{30}.\end{array}$(47)

and $\mathbb{R}{}^3$ spanned by

(48) $\begin{array}{cc}{e}_7+e_{10},& e_{26},e_{27}.\end{array}$(48)

We conclude that the symmetry Lie algebra is$\mathrm{s}(2,\mathrm{R})\times ({\mathrm{n}}_{24}\times {\mathrm{R}}^3)$.

6.12: $[e_1,e_3]=e_4,[e_1,e_4]=e_6,[e_2,e_5]=e_6$:

$S=\left[\begin{array}{ccccc}1& w& \frac{w^2}{2}& q& x\\ 0& 1& w& 0& y\\ 0& 0& 1& 0& z\\ 0& 0& 0& 1& p\\ 0& 0& 0& 0& 1\end{array}\right].$

Right-invariant vector fields: $-(D_w+y{D}_x+z{D}_y),D_p,D_z,D_y,D_q+p{D}_x,D_x$.

The geodesic equations are:

(49) $\begin{array}{c}\ddot{x}=\dot{p}\dot{q}+\dot{y}\dot{w}\\ \ddot{y}=\dot{z}\dot{w}\\ \ddot{z}=0\\ \ddot{w}=0\\ \ddot{p}=0\\ \ddot{q}=0.\end{array}$(49)

The symmetry algebra is spanned by

(50) $\begin{array}{cc}{e}_1& =D_q,e_2=D_z,e_3=D_w,e_4=t{D}_t,e_5=D_t,e_6=D_y,e_7=t{D}_x,e_8=D_x,\\ e_9& =D_y,e_{10}=p{D}_t,e_{11}=q{D}_t,e_{12}=w{D}_t,e_{13}=z{D}_t,e_{14}=z{D}_x,e_{15}=p{D}_x,\\ e_{16}& =q{D}_x,e_{17}=w{D}_x,e_{18}=t{D}_p+1/2tq{D}_x,e_{19}=1/2tw{D}_x+t{D}_y,e_{20}=q{D}_p+1/2q^2{D}_x,\\ e_{21}& =1/2w^2{D}_x+w{D}_y,e_{22}=t{D}_q+1/2tp{D}_x,e_{23}=p{D}_q+1/2p^2{D}_x,e_{24}=(wz-2y)D_t,\\ e_{25}& =q{D}_q+1/2pq{D}_x,e_{26}=w{D}_q+1/2pw{D}_x,e_{27}=z{D}_q+1/2zp{D}_x,e_{28}=p{D}_p+1/2pq{D}_x,\\ e_{29}& =1/2pw{D}_x+p{D}_y,e_{30}=1/2qw{D}_x+q{D}_y,e_{31}=1/2zw{D}_x+z{D}_y,e_{32}=w{D}_p+1/2qw{D}_x.\\ e_{33}& =z{D}_p+1/2zq{D}_x,e_{34}=(y-1/2zw)D_x,e_{35}=1/6w^3{D}_x+1/2w^2{D}_y+w{D}_z,\\ e_{36}& =(wz-2y)D_q+1/2p(wz-2y)D_x,e_{37}=(wz-2y)D_p+1/2q(wz-2y)D_x,\\ e_{38}& =-1/4w(wz-2y)D_x+(y-1/2zw)D_y,\\ e_{39}& =w{D}_w+(1/4z{w}^2-1/2yw-pq+2x)D_x+1/2zw{D}_y,\\ e_{40}& =(1/4z{w}^2-1/2yw-1/2pq+x)D_x+1/2zw{D}_y+z{D}_z.\end{array}$(50)

The symmetry algebra g is a 40-dimensional indecomposable Lie algebra. The Levi decomposition is given by $g=g_1\times {\mathrm{g}}_2$, where $g_1$ is $\mathrm{s}(4,\mathrm{R})$ and spanned by

(51) $\begin{array}{rl}& e_4-e_{38},e_{10},e_{11},e_{18},e_{19},e_{20},e_{22},e_{23},e_{24},e_{25}-e_{38},\\ & e_{28}-e_{38},e_{29},e_{30},e_{36},e_{37}.\end{array}$(51)

The solvable part ${\mathrm{g}}_2$ is a semidirect product of a 22-dimensional nilradical ${\mathrm{n}}_{22}$ spanned by

(52) $\begin{array}{cc}{e}_1,& e_2,e_3,e_5,e_6,e_7,e_8,e_9,e_{12},\\ e_{13}& e_{14},e_{15},e_{16},e_{17},e_{21},e_{26},e_{27},\\ e_{31},& e_{32},e_{33},e_{34},e_{35}\end{array}$(52)

and $\mathbb{R}{}^3$ spanned by

(53) $\begin{array}{cc}{e}_4+e_{25}+e_{28}+e_{38}& ,e_{39},e_{40}.\end{array}$(53)

We conclude that the symmetry Lie algebra is$\mathrm{s}(4,\mathrm{R})\times ({\mathrm{n}}_{22}\times {\mathrm{R}}^3)$.

6.13: $[e_1,e_2]=e_5,[e_1,e_3]=e_4,[e_1,e_4]=e_6,[e_2,e_5]=e_6$:

$S=\left[\begin{array}{ccccc}1& w& \frac{w^2}{2}& q& x\\ 0& 1& w& p& y\\ 0& 0& 1& 0& z\\ 0& 0& 0& 1& p\\ 0& 0& 0& 0& 1\end{array}\right].$

Right-invariant vector fields: $(D_w+p{D}_q+y{D}_x+z{D}_y),D_p+p{D}_y,-D_z,D_y,-D_q-p{D}_x,-D_x.$

The geodesic equations are:

(54) $\begin{array}{c}\ddot{x}=-2p\dot{p}\dot{w}+\dot{y}\dot{w}+\dot{p}\dot{q}-p{\dot{q}}^2\\ \ddot{y}=\dot{z}\dot{w}+{\dot{p}}^2\\ \ddot{z}=0\\ \ddot{w}=0\\ \ddot{p}=0\\ \ddot{q}=\dot{p}\dot{w}.\end{array}$(54)

The symmetry algebra is spanned by

(55) $\begin{array}{cc}{e}_1& =D_t,e_2=t{D}_p,e_3=D_p,e_4=D_x,e_5=D_y,e_6=D_q,e_7=D_z,e_8=D_w,\\ e_9& =t{D}_t,e_{10}=z{D}_p,e_{11}=w{D}_p,e_{12}=w{D}_t,e_{13}=z{D}_t,e_{14}=y{D}_p+z{D}_q,\\ e_{15}& =x{D}_p+y{D}_q+z{D}_x,e_{16}=tw{D}_p+2t{D}_q,e_{17}=q{D}_p+x{D}_q+y{D}_x+z{D}_y,\\ e_{18}& =\frac{w^2}{2}{D}_p+w{D}_q,e_{19}=p{D}_p+q{D}_q+x{D}_x+y{D}_y+z{D}_z,e_{20}=wz{D}_p+2D_q,e_{21}=(wz-2y)D_t,\\ e_{22}& =4p{D}_p+3q{D}_q+w{D}_w+2x{D}_x+y{D}_y,e_{23}=\frac{w^3}{6}{D}_p+\frac{w^2}{2}{D}_q+w{D}_x,\\ e_{24}& =\frac{w^4}{24}{D}_p+\frac{w^3}{6}{D}_q+\frac{w^2}{2}x+w{D}_y,e_{25}=(-\frac{w^{2}z}{2}+yw)D_p+(2y-wz)D_q,\\ e_{26}& =\frac{w^5}{120}{D}_p+\frac{w^4}{24}{D}_q+\frac{w^3}{6}{D}_x+\frac{w^2}{2}{D}_y+w{D}_z,\\ e_{27}& =1/6w(w^{2}z-3wy+6x)D_p+(2x-wy+1/2z{w}^2)D_q+wz{D}_x+2z{D}_y\\ e_{28}& =(-\frac{z{w}^4}{24}+\frac{y{w}^3}{6}-\frac{x{w}^2}{2}+qw)D_p+(2q-wx+\frac{w^{2}y}{2}-\frac{w^{3}z}{6})D_q+(-\frac{z{w}^2}{2}+wy)D_x+(-wz+2y)D_y.\end{array}$(55)

The symmetry algebra g is a 28-dimensional indecomposable solvable Lie algebra. It has a 24-dimensional radical ${\mathrm{n}}_{24}$ spanned by

(56) $\begin{array}{cc}{e}_1,& e_2,e_3,e_4,e_5,e_6,e_7,e_8,e_{10},e_{11},e_{12},e_{13},e_{14},\\ e_{15},& e_{16},e_{17},e_{18},e_{20},e_{21},e_{23},e_{24},e_{25},e_{26},e_{27},\end{array}$(56)

and a 4-dimensional abelian complement $R{}^4$ spanned by

(57) $\begin{array}{cc}{e}_9,& e_{19},e_{22},e_{28},\end{array}$(57)

We conclude that the symmetry Lie algebra is${\mathrm{n}}_{24}\times {\mathrm{R}}^4$.

6.14: $[e_1,e_3]=e_4,[e_1,e_4]=e_6,[e_2,e_3]=e_5,[e_2,e_5]=a{e}_6$: $a$ can be reduced to $a=\pm 1$.

$a=1:$

$S=\left[\begin{array}{ccccc}1& p& q& \frac{1}{2}(p^2+q^2)& x\\ 0& 1& 0& p& y\\ 0& 0& 1& q& z\\ 0& 0& 0& 1& w\\ 0& 0& 0& 0& 1\end{array}\right]$

Right-invariant vector fields: $D_p+y{D}_x+w{D}_y,D_q+z{D}_x+w{D}_z,D_w,-D_y,-D_z,D_x$.

The geodesic equations are:

(58) $\begin{array}{c}\ddot{x}=\dot{p}\dot{y}+\dot{q}\dot{z}\\ \ddot{y}=\dot{p}\dot{w}\\ \ddot{z}=\dot{q}\dot{w}\\ \ddot{w}=0\\ \ddot{p}=0\\ \ddot{q}=0.\end{array}$(58)

The symmetry algebra is spanned by

(59) $\begin{array}{cc}{e}_1& =D_t,e_2=D_y,e_3=D_z,e_4=t{D}_x,e_5=D_p,e_6=D_x,e_7=D_q,e_8=D_w,\\ e_9& =t{D}_t,e_{10}=q{D}_t,e_{11}=w{D}_t,e_{12}=w{D}_x,e_{13}=p{D}_x,e_{14}=q{D}_x,e_{15}=p{D}_t,\\ e_{16}& =y{D}_x+w{D}_y,e_{17}=z{D}_x+w{D}_z,e_{18}=w{D}_w+x{D}_x+y{D}_y+z{D}_z,\\ e_{19}& =1/2tp{D}_x+t{D}_y,e_{20}=1/2p^2{D}_x+p{D}_y,e_{21}=1/2tq{D}_x+t{D}_z,e_{22}=1/2q^2{D}_x+q{D}_z,\\ e_{23}& =q{D}_p-p{D}_q+z{D}_y-y{D}_z,e_{24}=p{D}_p+q{D}_q+2x{D}_x+y{D}_y+z{D}_z,e_{25}=(qw-2z)D_t,\\ e_{26}& =(pw-2y)D_x,e_{27}=(qw-2z)D_x,e_{28}=(pw-2y)D_t,e_{29}=1/2qp{D}_x+q{D}_y,\\ e_{30}& =1/2qp{D}_x+p{D}_z,e_{31}=(1/2w{p}^2-yp)D_x+(pw-2y)D_y,\\ e_{32}& =(1/2w{q}^2-zq)D_x+(qw-2z)D_z,\\ e_{33}& =(1/2qw-z)p{D}_x+(qw-2z)D_y,e_{34}=1/2q(pw-2y)D_x+(pw-2y)D_z.\end{array}$(59)

The symmetry algebra $\mathfrak{g}$ is a 34-dimensional indecomposable Lie algebra. The Levi decomposition is given by $\mathfrak{g}={\mathfrak{g}}_1{\mathfrak{g}}_2$, where ${\mathfrak{g}}_1$ is $\mathfrak{s}\mathfrak{l}(3,\mathbb{R})$ and spanned by

(60) $\begin{array}{cc}{e}_9+\frac{1}{2}{e}_{32},& e_{19},e_{21},e_{25},e_{28},e_{31}-e_{e32},e_{33},e_{34}.\end{array}$(60)

The solvable part ${\mathrm{g}}_2$ is a semidirect product of a 22-dimensional nilradical ${\mathrm{n}}_{22}$ spanned by

(61) $\begin{array}{cc}{e}_1,& e_2,e_3,e_4,e_5,e_6,e_7,e_8,\\ e_{10},& e_{11},e_{12},e_{13},e_{14},e_{15},e_{16},\\ e_{17},& e_{20},e_{22},e_{26},e_{27},e_{29},e_{30}\end{array}$(61)

and $\mathbb{R}{}^4$ spanned by

(62) $\begin{array}{cc}{e}_9-\frac{1}{2}(e_{31}+e_{32})& ,e_{18}+\frac{1}{2}(e_{31}+e_{32}),e_{23}+\frac{1}{2}(e_{33}-e_{34}),e_{24}+\frac{1}{2}(e_{31}+e_{32}).\end{array}$(62)

We conclude that the symmetry Lie algebra is $(\mathfrak{s}\mathfrak{l}(3,\mathbb{R})({\mathfrak{n}}_{22}{\mathbb{R}}^4)$.

$a=-1$

$S=\left[\begin{array}{cccc}1& x& y& z\\ 0& 1& w& p\\ 0& 0& 1& q\\ 0& 0& 0& 1\end{array}\right].$

Right-invariant vector fields: $\frac{1}{2}(D_q+D_x+w{D}_y+p{D}_z),\frac{1}{2}(D_x-D_q+w{D}_y+p{D}_z),-2D_w-2q{D}_p,D_y-D_p+q{D}_z,D_p+D_y+q{D}_z,D_z$.

The geodesic equations are:

(63) $\begin{array}{c}\ddot{x}=0\\ \ddot{y}=\dot{x}\dot{w}\\ \ddot{z}=-w\dot{q}\dot{x}+\dot{p}\dot{x}+\dot{y}\dot{q}\\ \ddot{w}=0\\ \ddot{p}=\dot{q}\dot{w}\\ \ddot{q}=0.\end{array}$(63)

The symmetry algebra is spanned by

(64) $\begin{array}{cc}{e}_1& =D_t,e_2=D_p,e_3=D_y,e_4=t{D}_z,e_5=D_z,e_6=D_q,e_7=D_x,e_8=t{D}_t,\\ e_9& =w{D}_t,e_{10}=x{D}_t,e_{11}=w{D}_z,e_{12}=q{D}_z,e_{13}=x{D}_z,e_{14}=q{D}_t,e_{15}=w{D}_p+y{D}_z,\\ e_{16}& =w{D}_y+p{D}_z,e_{17}=p{D}_p+q{D}_q+z{D}_z,e_{18}=x{D}_x+y{D}_y+z{D}_z,\\ e_{19}& =p{D}_p+w{D}_w+y{D}_y+z{D}_z,e_{20}=t{D}_p+1/2tx{D}_z,e_{21}=x{D}_p+1/2x^2{D}_z,e_{22}=t{D}_y+1/2tq{D}_z,\\ e_{23}& =q{D}_y+1/2q^2{D}_z,e_{24}=D_w-1/2qx{D}_z,e_{25}=(wx-2y)D_t,\\ e_{26}& =(qw-2p)D_z,e_{27}=(wx-2y)D_z,e_{28}=(qw-2p)D_t,e_{29}=q{D}_p+1/2qx{D}_z,\\ e_{30}& =x{D}_y+1/2qx{D}_z,e_{31}=(wx-2y)D_p+(1/2w{x}^2-xy)D_z,\\ e_{32}& =(qw-2p)D_y+(1/2w{q}^2-pq)D_z,\\ e_{33}& =(qw-2p)D_p+(1/2qw-p)x{D}_z,e_{34}=(wx-2y)D_y+(1/2wx-y)q{D}_z.\end{array}$(64)

The symmetry algebra g is a 34-dimensional indecomposable Lie algebra. The Levi decomposition$g=g_1\times {\mathrm{g}}_2$, where $g_1$ is $\mathrm{s}(3,\mathrm{R})$ and spanned by

(65) $\begin{array}{cc}{e}_8+\frac{1}{2}{e}_{34},& e_{20},e_{22},e_{25},e_{28},e_{31},e_{32},e_{33}-e_{34}.\end{array}$(65)

The solvable part ${\mathrm{g}}_2$ is a semidirect product of a 22-dimensional nilradical ${\mathrm{n}}_{22}$ spanned by

(66) $\begin{array}{cc}{e}_1,& e_2,e_3,e_4,e_5,e_6,e_7,e_9,\\ e_{10},& e_{11},e_{12},e_{13},e_{14},e_{15},e_{16},\\ e_{21},& e_{23},e_{24},e_{26},e_{27},e_{29},e_{30}\end{array}$(66)

and $\mathbb{R}{}^4$ spanned by

(67) $\begin{array}{cc}{e}_8-\frac{1}{2}(e_{33}+e_{34})& ,e_{17}+\frac{1}{2}{e}_{33},e_{18}+\frac{1}{2}{e}_{34},e_{19}+\frac{1}{2}(e_{33}+e_{34}).\end{array}$(67)

We conclude that the symmetry Lie algebra is$\mathrm{s}(3,\mathrm{R})\times ({\mathrm{n}}_{22}\times {\mathrm{R}}^4)$.

6.15: $[e_1,e_2]=e_3+e_5,[e_1,e_3]=e_4,[e_1,e_4]=e_6,[e_2,e_5]=e_6$:

$S=\left[\begin{array}{cccccc}1& 2w& w^2-z& p+y-zw+\frac{w^3}{3}& q& \hspace{0.17em}\\ 0& 1& w& \frac{w^2}{2}& x& \hspace{0.17em}\\ 0& 0& 1& w& y& \hspace{0.17em}\\ 0& 0& 0& 1& z& \hspace{0.17em}\\ 0& 0& 0& 0& 1& \hspace{0.17em}\end{array}\right].$

Right-invariant vector fields are: $D_w+z{D}_y+y{D}_x+2x{D}_q,D_z-y{D}_q,-D_y+z{D}_q+2D_p,D_x,-2D_p-$ $2z{D}_q,-2D_q$.

The geodesic equations are:

(68) $\begin{array}{c}\ddot{x}=\dot{y}\dot{w}\\ \ddot{y}=\dot{z}\dot{w}\\ \ddot{z}=0\\ \ddot{w}=0\\ \ddot{p}=0\\ \ddot{q}=\dot{z}\dot{p}+2\dot{x}\dot{w}-z\dot{z}\dot{w}.\end{array}$(68)

The symmetry algebra is spanned by

(69) $\begin{array}{cc}{e}_1& =D_w,e_2=t{D}_t,e_3=D_t,e_4=D_x,e_5=D_y,e_6=t{D}_q,e_7=D_q,e_8=D_p,\\ e_9& =p{D}_t,e_{10}=w{D}_t,e_{11}=z{D}_t,e_{12}=p{D}_q,e_{13}=z{D}_q,e_{14}=w{D}_q,e_{15}=tw{D}_q+t{D}_x,\\ e_{16}& =w^2{D}_q+w{D}_x,e_{17}=2x{D}_q+y{D}_x+z{D}_y,e_{18}=t{D}_p+1/2tz{D}_q,\\ e_{19}& =pw{D}_q+p{D}_x,e_{20}=wz{D}_q+z{D}_x,e_{21}=z{D}_p+1/2z^2{D}_q,e_{22}=-1/2wz{D}_q+D_z,\\ e_{23}& =(wz-2y)D_t,e_{24}=p{D}_p+1/2pz{D}_q,e_{25}=w{D}_p+1/2wz{D}_q,\\ e_{26}& =(-1/2wz+y)D_q,e_{27}=1/3w^3{D}_q+1/2w^2{D}_x+w{D}_y,\\ e_{28}& =(wz-2y)D_p+(1/2w{z}^2-yz)D_q,\\ e_{29}& =(w^{2}z-2wy+4x)D_q+wz{D}_x+2z{D}_y,\\ e_{30}& =(1/12w^4-1/2z{w}^2+(1/2p+y)w-2x)D_q+1/6w^3{D}_x+1/2w^2{D}_y+w{D}_z,\\ e_{31}& =(1/3z{w}^3-y{w}^2+2xw-1/2pz+q)D_q+w{D}_w+(1/2z{w}^2-wy+2x)D_x+(wz-y)D_y,\\ e_{32}& =(-1/6z{w}^3+1/2y{w}^2-xw-1/2pz+2q)D_q\\ \hspace{0.17em}& +(-1/4z{w}^2+1/2yw+x)D_x+(-1/2wz+2y)D_y+z{D}_z.\end{array}$(69)

The symmetry algebra g is a 32-dimensional indecomposable Lie algebra. The Levi decomposition is given by $g=g_1\times {\mathrm{g}}_2$, where $g_1$ is $\mathrm{s}(2,\mathrm{R})$ and spanned by

(70) $\begin{array}{cc}{e}_2-e_4,& e_9,e_{18}.\end{array}$(70)

The solvable part ${\mathrm{g}}_2$ is 29-dimensional with a 26-dimensional nilradical ${\mathrm{n}}_{26}$ spanned by

(71) $\begin{array}{cc}{e}_1,& e_3,e_4,e_5,e_6,e_7,e_8,e_{10},\\ e_{11},& e_{12},e_{13},e_{14},e_{15},e_{16},e_{17},\\ e_{19},& e_{20},e_{21},e_{22},e_{23},e_{25},e_{36},\\ e_{27},& e_{28},e_{29},e_{30}.\end{array}$(71)

We conclude that the symmetry Lie algebra is$\mathrm{s}(2,\mathrm{R})\times ({\mathrm{n}}_{26}\times R^3)$.

6.16: $[e_1,e_3]=e_4,[e_1,e_4]=e_5,[e_1,e_5]=e_6,[e_2,e_3]=e_5,[e_2,e_4]=e_6$:

$S=\left[\begin{array}{ccccc}1& w& p+\frac{w^2}{2}& pw+\frac{w^3}{6}& q\\ 0& 1& w& p+\frac{w^2}{2}& x\\ 0& 0& 1& w& y\\ 0& 0& 0& 1& z\\ 0& 0& 0& 0& 1\end{array}\right].$

Right-invariant vector fields: $-(D_w+x{D}_q+y{D}_x+z{D}_y),-(D_p+y{D}_q+z{D}_x),D_z,D_y,D_x,D_q$.

The geodesic equations are:

(72) $\begin{array}{c}\ddot{x}=pz\dot{y}\dot{w}\\ \ddot{y}=\dot{z}\dot{w}\\ \ddot{z}=0\\ \ddot{w}=0\\ \ddot{p}=0\\ \ddot{q}=\dot{y}\dot{p}+\dot{x}\dot{w}.\end{array}$(72)

The symmetry algebra is spanned by

(73) $\begin{array}{cc}{e}_1& =D_t,e_2=D_y,e_3=t{D}_q,e_4=D_q,e_5=D_x,e_6=D_p,e_7=D_z,e_8=D_w,\\ e_9& =t{D}_t,e_{10}=p{D}_t,e_{11}=w{D}_t,e_{12}=z{D}_t,e_{13}=z{D}_q,e_{14}=p{D}_q,e_{15}=w{D}_q,\\ e_{16}& =y{D}_q+z{D}_x,e_{17}=x{D}_q+y{D}_x+z{D}_y,e_{18}=tw{D}_q+2t{D}_x,\\ e_{19}& =q{D}_q+x{D}_x+y{D}_y+z{D}_z,e_{20}=w^2/2D_q+w{D}_x,e_{21}=wz{D}_q+2z{D}_x,e_{22}=pw{D}_q+2p{D}_x,\\ e_{23}& =(wz-2y)D_t,e_{24}=w{D}_p+(-pz+2x)D_q+y{D}_x,e_{25}=1/6w^3{D}_q+(1/2w^2-p)D_x+w{D}_y,\\ e_{26}& =(-1/2w^{2}z+wy)D_q+(-wz+2y)D_x,\\ e_{27}& =(1/24w^4+1/2p{w}^2+1/2p^2)D_q+(1/6w^3+pw)D_x+(1/2w^2+p)D_y+w{D}_z,\\ e_{28}& =(1/6z{w}^3-1/2w^{2}y+xw-yp+q)D_q+w{D}_w+((1/2w^2-p)z-wy+2x)D_x+(wz-y)D_y,\\ e_{29}& =p{D}_p+(-1/2xw+q+1/2yp+1/4w^{2}y-1/12z{w}^3)D_q\\ \hspace{0.17em}& +((-1/4w^2+1/2p)z+1/2wy)D_x+(y-1/2zw)D_y.\end{array}$(73)

The symmetry algebra g is a 29-dimensional indecomposable solvable Lie algebra. It has a 25-dimensional nilradical ${\mathrm{n}}_{25}$ spanned by

(74) $\begin{array}{cc}{e}_1,& e_2,e_3,e_4,e_5,e_6,e_7,e_8,e_{10},e_{11},e_{12},e_{13},e_{14},\\ e_{15},& e_{16},e_{17},e_{18},e_{20},e_{21},e_{22},e_{23},e_{24},e_{25},e_{26},e_{27},\end{array}$(74)

and a 4-dimensional abelian complement $R{}^4$ spanned by

(75) $\begin{array}{cc}{e}_9,& e_{19},e_{28},e_{29}.\end{array}$(75)

We conclude that the symmetry Lie algebra is${\mathrm{n}}_{25}\times {\mathrm{R}}^4$.

6.17: $[e_1,e_2]=e_3,[e_1,e_3]=e_4,[e_1,e_4]=e_6,[e_2,e_5]=e_6$:

$S=\left[\begin{array}{cccccc}1& w& \frac{w^2}{2}& p+\frac{w^3}{6}& q& \hspace{0.17em}\\ 0& 1& w& \frac{w^2}{2}& x& \hspace{0.17em}\\ 0& 0& 1& w& y& \hspace{0.17em}\\ 0& 0& 0& 1& z& \hspace{0.17em}\\ 0& 0& 0& 0& 1& \hspace{0.17em}\end{array}\right].$

Right-invariant vector fields: $-(D_w+x{D}_q+y{D}_x+z{D}_y),D_z,D_y,D_x,D_p+z{D}_q,D_q$.

The geodesic equations are:

(76) $\begin{array}{c}\ddot{x}=\dot{y}\dot{w}\\ \ddot{y}=\dot{z}\dot{w}\\ \ddot{z}=0\\ \ddot{w}=0\\ \ddot{p}=0\\ \ddot{q}=\dot{z}\dot{p}+\dot{x}\dot{w}.\end{array}$(76)

The symmetry algebra is spanned by

(77) $\begin{array}{cc}{e}_1& =D_z,e_2=D_w,e_3=t{D}_t,e_4=D_t,e_5=D_y,e_6=D_x,e_7=t{D}_q,e_8=D_q,\\ e_9& =D_p,e_{10}=p{D}_t,e_{11}=w{D}_t,e_{12}=z{D}_t,e_{13}=p{D}_q,e_{14}=z{D}_q,e_{15}=w{D}_q,\\ e_{16}& =t{D}_p+1/2tz{D}_q,e_{17}=1/2tw{D}_q+t{D}_x,e_{18}=1/2w^2{D}_q+w{D}_x,\\ e_{19}& =z{D}_p+1/2z^2{D}_q,e_{20}=(wz-2y)D_t,e_{21}=p{D}_p+1/2pz{D}_q,e_{22}=1/2pw{D}_q+p{D}_x,\\ e_{23}& =1/2wz{D}_q+z{D}_x,e_{24}=w{D}_p+1/2wz{D}_q,e_{25}=(-1/2wz+y)D_q,\\ e_{26}& =1/6w^3{D}_q+1/2w^2{D}_x+w{D}_y,e_{27}=(wz-2y)D_p+(1/2z^{2}w-yz)D_q,\\ e_{28}& =(1/24w^4+1/2pw)D_q+1/6w^3{D}_x+1/2w^2{D}_y+w{D}_z,\\ e_{29}& =-1/4w(wz-2y)D_q+(-1/2wz+y)D_x,\\ e_{30}& =(1/4z{w}^2-1/2yw+x)D_q+1/2wz{D}_x+z{D}_y,\\ e_{31}& =(1/12z{w}^3-1/4y{w}^2+1/2xw+1/2pz)D_q+(1/4z{w}^2-1/2yw+x)D_x+1/2wz{D}_y+z{D}_z,\\ e_{32}& =(-1/12z{w}^3+1/4y{w}^2-1/2xw-1/2pz+q)D_q-1/4w(wz-2y)D_x+(-1/2wz+y)D_y,\\ e_{33}& =(1/12z{w}^3-1/4y\ast w^2+1/2xw-pz+2q)D_q+w{D}_w\\ \hspace{0.17em}& +(2x+1/4z{w}^2-1/2yw)D_x+1/2wz{D}_y\\ e_{34}& =(1/24z{w}^4-1/6y{w}^3+1/2x{w}^2+pwz-qw-yp)D_q\\ \hspace{0.17em}& +(1/6z{w}^3-1/2y{w}^2+xw+pz-2q)D_x+(1/2z{w}^2-wy)D_y+(wz-2y)D_z.\end{array}$(77)

The symmetry algebra g is a 34-dimensional indecomposable Lie algebra. The Levi decomposition is given by $g=g_1\times {\mathrm{g}}_2$, where $g_1$ is $\mathrm{s}(2,\mathrm{R})\oplus \mathrm{s}(2,\mathrm{R})$ and spanned by

(78) $\begin{array}{cc}{e}_3-e_{21},& e_{10},e_{16},e_{30},e_{31}-e_{32},e_{34}.\end{array}$(78)

The solvable part ${\mathrm{g}}_2$ is a semidirect product of a 25-dimensional nilradical ${\mathrm{n}}_{25}$ spanned by

(79) $\begin{array}{cc}{e}_1,& e_2+\frac{1}{2}{e}_{30},e_4,e_5,e_6,e_7,e_8,e_9,\\ e_{11}& e_{12},e_{13},e_{14},e_{15},e_{17},e_{18},e_{19},\\ e_{20},& e_{22},e_{23},e_{24},e_{25},e_{26},e_{27},e_{28},e_{29}\end{array}$(79)

and $\mathbb{R}{}^3$ spanned by

(80) $\begin{array}{cc}{e}_3+e_{21}& ,e_{31}+e_{32},e_{33}.\end{array}$(80)

We conclude that the symmetry Lie algebra is$(\mathrm{s}(2,\mathrm{R})\oplus \mathrm{s}(2,\mathrm{R}))\times ({\mathrm{n}}_{25}\times {\mathrm{R}}^3)$.

6.18: $[e_1,e_2]=e_3,[e_1,e_3]=e_4,[e_1,e_4]=e_6,[e_2,e_3]=e_5,[e_2,e_5]=a{e}_6$. As in 6.5, 6.10 and 6.14 $a$ can be reduced to $a=\pm 1$.

$a=1:$

$S=\left[\begin{array}{ccccc}1& y+3z& \frac{y^2}{2}+5yz+\frac{15z^2}{2}-2x& w& p\\ 0& 1& y+5z& yz+\frac{5z^2}{2}-x& q\\ 0& 0& 1& z& x\\ 0& 0& 0& 1& y\\ 0& 0& 0& 0& 1\end{array}\right].$

Right-invariant vector fields: $D_y+(\frac{5}{2}{z}^2+yz-x)D_w+q{D}_p+x{D}_q,D_z+y{D}_x+(3yz+\frac{15}{2}{z}^2-3x)D_w+$ $3q{D}_p+5x{D}_q,D_x-2z{D}_w-2x{D}_p-y{D}_q,D_q,D_p,D_w+y{D}_p$.

The geodesic equations are:

(81) $\begin{array}{c}\ddot{x}=\dot{y}\dot{z}\\ \ddot{y}=0\\ \ddot{z}=0\\ \ddot{w}=(5y+15z){\dot{z}}^2-(5\dot{x}-(y+8z)\dot{y})\dot{z}+z{\dot{y}}^2-\dot{x}\dot{y}\\ \ddot{p}=1/2(-2yz-5z^2+2x){\dot{y}}^2+1/2((-10yz-15z^2+6x)\dot{z}+4z\dot{x}+2\dot{q}+2\dot{w})\dot{y}-2{\dot{x}}^2+3\dot{z}\dot{q}\\ \ddot{q}=(y\dot{y}+5\dot{x})\dot{z}.\end{array}$(81)

The symmetry algebra is spanned by

(82) $\begin{array}{cc}{e}_1& =t{D}_t,e_2=D_w,e_3=D_q,e_4=t{D}_p,e_5=D_p,e_6=D_t,e_7=y{D}_p,e_8=z{D}_p,\\ e_9& =y{D}_t,e_{10}=z{D}_t,e_{11}=x{D}_p+z{D}_w,e_{12}=1/2ty{D}_p+t{D}_w,e_{13}=1/2y^2{D}_p+y{D}_w,\\ e_{14}& =(1/2y^2+3x)D_p+D_x,e_{15}=(yz-2x)D_t,\\ e_{16}& =(yz-2x)D_p,e_{17}=(1/2y^2+3x)D_p+y{D}_q,e_{18}=(3/2z^2+x)D_p+z{D}_q,\\ e_{19}& =1/2t(y+3z)D_p+t{D}_q,e_{20}=4p{D}_p+3q{D}_q+3w{D}_w+2x{D}_x+y{D}_y+z{D}_z,\\ e_{21}& =(1/2y^{2}z-yx)D_p+(yz-2x)D_w,\\ e_{22}& =(-5/2y^{2}z+5yx)D_p+(8x+15/2z^2+1/2y^2)D_w+D_z,\\ e_{23}& =(-y-3z)(-1/2yz+x)D_p+(yz-2x)D_q,\\ e_{24}& =(10/3z^3+10/3y{z}^2+(1/6y^2-8/3x)z+1/3yx+\\ \hspace{0.17em}& 1/3q-4/3w)D_p+D_{q}x+(x+5/2z^2)D_w+D_y\\ e_{25}& =(-50/3z^3-50/3y{z}^2+(-7/3y^2+40/3x)z+4/3yx\\ \hspace{0.17em}& +13/3q+20/3w)D_p+5x{D}_q+(-5x-1/2y^2)D_w+y{D}_x,\\ e_{26}& =(40/3z^3+65/6y{z}^2+(8/3y^2-11/3x)z-8/3yx-11/3q-13/3w)D_p\\ \hspace{0.17em}& +5/2z^2{D}_q+(-x-5/2z^2)D_w+z{D}_x.\end{array}$(82)

The symmetry algebra g is a 26-dimensional indecomposable solvable Lie algebra. It has a 24-dimensional nilradical ${\mathrm{n}}_{24}$ spanned by

(83) $\begin{array}{cc}{e}_2,& e_3,e_4,e_5,e_6,e_7,e_8,e_9,e_{10},e_{11},e_{12},e_{13},e_{14},\\ e_{15},& e_{16},e_{17},e_{18},e_{19},e_{21},e_{22},e_{23},e_{24},e_{25},e_{26},\end{array}$(83)

and a two-dimensional abelian complement $R{}^2$ spanned by

(84) $\begin{array}{cc}{e}_1,& e_{20}.\end{array}$(84)

We conclude that the symmetry Lie algebra is${\mathrm{n}}_{24}\times {\mathrm{R}}^2$.

$a=-1:$

$S=\left[\begin{array}{ccccc}1& p& x& y& w\\ 0& 1& q& pq-x& z\\ 0& 0& 1& p& x\\ 0& 0& 0& 1& q\\ 0& 0& 0& 0& 1\end{array}\right]$

Right-invariant vector fields: $\begin{array}{rl}& \frac{1}{2}(-D_p+2D_q-\left(q-2\right)D_x+\left(2p+x-pq\right)D_y+2\left(x-q\right)D_z+\left(2x-z\right)D_w),\frac{1}{2}(D_p+\\ & 2D_q+\left(q+2\right)D_x+\left(pq+2p-x\right)D_y+2\left(x-q\right)D_z+\left(2x+z\right)D_w),D_x+\left(p-2\right)D_y-q{D}_z+\left(x-2q\right)\\ & D_w,-D\mathrm{\_}y-2D_z-\left(q+2\right)D_w,D_y-2D_z+\left(q-2\right)D_w,-2D_w\end{array}$.

The geodesic equations are:

(85) $\begin{array}{c}\ddot{x}=\dot{p}\dot{q}\\ \ddot{y}=p\dot{p}\dot{q}\\ \ddot{z}=-q\dot{p}\dot{q}\\ \ddot{w}=-q\dot{x}\dot{p}-p\dot{x}\dot{q}+{\dot{x}}^2+(x-2q)\dot{p}\dot{q}+\dot{q}\dot{y}+\frac{1}{2}\dot{z}\dot{q}\\ \ddot{p}=0\\ \ddot{q}=0.\end{array}$(85)

The symmetry algebra is spanned by

(86) $\begin{array}{cc}{e}_1& =D_y,e_2=D_z,e_3=t{D}_w,e_4=D_w,e_5=t{D}_t,e_6=D_t,e_7=p{D}_w,e_8=q{D}_w,\\ e_9& =p{D}_t,e_{10}=q{D}_t,e_{11}=x{D}_y+D_p,e_{12}=1/2q^2{D}_w+q{D}_y,e_{13}=1/4tq{D}_w+t{D}_z,\\ e_{14}& =1/4q^2{D}_w+q{D}_z,e_{15}=1/2tq{D}_w+t{D}_y,\\ e_{16}& =1/2pq{D}_w+D_x,e_{17}=(pq-2x)D_t,e_{18}=1/2pq{D}_w+p{D}_y,\\ e_{19}& =1/4pq{D}_w+p{D}_z,e_{20}(-1/2pq+x)D_w,\\ e_{21}& =p{D}_p+q{D}_q+(-2z+4w)D_w+2x{D}_x+3y{D}_y+3z{D}_z,\\ e_{22}& =(-1/2p^{2}q+xp+y)D_w+p{D}_x,\\ e_{23}& =(-1/2p{q}^2+xq-z)D_w+q{D}_x,\\ e_{24}& =(-xq+1/2p{q}^2)D_w+(pq-2x)D_y\\ e_{25}& =1/4q(pq-2x)D_w+(pq-2x)D_z,\\ e_{26}& =D_q+((-q-1/2x)p-1/4xq+1/2y-1/4z)D_w-x{D}_z.\end{array}$(86)

The symmetry algebra g is a 26-dimensional indecomposable solvable Lie algebra. It has a 24-dimensional nilradical ${\mathrm{n}}_{24}$ spanned by

(87) $\begin{array}{cc}{e}_1,& e_2,e_3,e_4,e_6,e_7,e_8,e_9,e_{10},e_{11},e_{12},e_{13},e_{14},\\ e_{15},& e_{16},e_{17},e_{18},e_{19},e_{20},e_{22},e_{23},e_{24},e_{25},e_{26},\end{array}$(87)

and a two-dimensional abelian complement $R{}^2$ spanned by

(88) $\begin{array}{cc}{e}_5,& e_{21}.\end{array}$(88)

We conclude that the symmetry Lie algebra is${\mathrm{n}}_{24}\times {\mathrm{R}}^2$.

6.19: $[e_1,e_2]=e_3,[e_1,e_3]=e_4,[e_1,e_4]=e_5,[e_1,e_5]=e_6,[e_2,e_3]=e_6$:

$S=\left[\begin{array}{cccccc}1& w& \frac{w^2}{2}& z+\frac{w^3}{6}& \frac{w^4}{24}-y+zw& p\\ 0& 1& w& \frac{w^2}{2}& \frac{w^3}{6}& q\\ 0& 0& 1& w& \frac{w^2}{2}& x\\ 0& 0& 0& 1& w& y\\ 0& 0& 0& 0& 1& z\\ 0& 0& 0& 0& 0& 1\end{array}\right].$

Right-invariant vector fields: $D_w+q{D}_p+x{D}_q+y{D}_x+z{D}_y,\frac{1}{2}(D_z+y{D}_p),-\frac{1}{2}(D_y-z{D}_p),\frac{1}{2}{D}_x,-\frac{1}{2}{D}_q,\frac{1}{2}{D}_p$.

The geodesic equations are

(89) $\begin{array}{c}\ddot{x}=\dot{y}\dot{w}\\ \ddot{y}=\dot{z}\dot{w}\\ \ddot{z}=0\\ \ddot{w}=0\\ \ddot{p}=z\dot{z}\dot{w}+\dot{q}\dot{w}\\ \ddot{q}=\dot{x}\dot{w}.\end{array}$(89)

The symmetry algebra is spanned by

(90) $\begin{array}{cc}{e}_1& =D_q,e_2=D_x,e_3=D_y,e_4=t{D}_p,e_5=D_p,e_6=D_w,e_7=t{D}_t,e_8=D_t,\\ e_9& =z{D}_p,e_{10}=w{D}_p,e_{11}=w{D}_t,e_{12}=z{D}_t,e_{13}=x{D}_p+y{D}_q+z{D}_x,\\ e_{14}& =q{D}_p+x{D}_q+y{D}_x+z{D}_y,e_{15}=1/2tw{D}_p+t{D}_q\\ e_{16}& =1/2w^2{D}_p+w{D}_q,e_{17}=1/2wz{D}_p+D_z,e_{18}=(wz-2y)D_t,\\ e_{19}& =1/2wz{D}_p+z{D}_q,e_{20}=(-1/2wz+y)D_p,\\ e_{21}& =1/6w^3{D}_p+1/2w^2{D}_q+w{D}_x,\\ e_{22}& =1/24w^4{D}_p+1/6w^3{D}_q+1/2w^2{D}_x+w{D}_y,\\ e_{23}& =7/3p{D}_p+2q{D}_q+1/3w{D}_w+5/3x{D}_x+4/3y{D}_y+z{D}_z,\\ e_{24}& =(1/2w^{2}z-wy+2x)D_p+wz{D}_q+2z{D}_x\\ e_{25}& =(-wy+1/120w^5+2x+1/2w^{2}z)D_p+1/24w^4{D}_q+1/6w^3{D}_x+1/2w^2{D}_y+w{D}_z,\\ e_{26}& =(-1/6z{w}^3+1/2w^{2}y-xw+2q)D_p+(-1/2w^{2}z+wy)D_q+(-wz+2y)D_x\\ e_{27}& =(1/24z{w}^4-1/6y{w}^3+1/2x{w}^2-qw)D_p+(1/6z{w}^3-1/2w^{2}y+xw-2q)D_q\\ \hspace{0.17em}& +(1/2w^{2}z-wy)D_x+(wz-2y)D_y.\end{array}$(90)

The symmetry algebra g is a 27-dimensional indecomposable solvable Lie algebra. It has a 24-dimensional nilradical ${\mathrm{n}}_{24}$ spanned by

(91) $\begin{array}{cc}{e}_1,& e_2,e_3,e_4,e_5,e_6,e_8,e_9,e_{10},e_{11},e_{12},e_{13},e_{14},\\ e_{15},& e_{16},e_{17},e_{18},e_{19},e_{20},e_{21},e_{22},e_{24},e_{25},e_{26}.\end{array}$(91)

and a three-dimensional abelian complement $\mathbb{R}{}^3$ spanned by

(92) $\begin{array}{cc}{e}_7,& e_{23},e_{27}.\end{array}$(92)

We conclude that the symmetry Lie algebra is ${\mathrm{n}}_{24}\times {\mathrm{R}}^3$.

6.20: $[e_1,e_2]=e_3,[e_1,e_3]=e_4,[e_1,e_4]=e_5,[e_1,e_5]=e_6,[e_2,e_3]=e_5,[e_2,e_4]=e_6$:

$S=\left[\begin{array}{cccccc}1& w& z+w^2& zw-2y+\frac{w^3}{3}& 3x-2yw+\frac{w^{2}z}{2}+\frac{w^4}{12}& p\\ 0& 1& 2w& w^2-z& y-zw+\frac{w^3}{3}& q\\ 0& 0& 1& w& \frac{w^2}{2}& x\\ 0& 0& 0& 1& w& y\\ 0& 0& 0& 0& 1& z\\ 0& 0& 0& 0& 0& 1\end{array}\right].$

Right-invariant vector fields: $D_w+q{D}_p+2x{D}_q+y{D}_x+z{D}_y,D_z+x{D}_p-y{D}_q,-(D_y-2y{D}_p+$ $z{D}_q),D_x+3z{D}_p,-2D_q,2D_p$.

The geodesic equations are:

(93) $\begin{array}{c}\ddot{x}=\dot{y}\dot{w}\\ \ddot{y}=\dot{z}\dot{w}\\ \ddot{z}=0\\ \ddot{w}=0\\ \ddot{p}=4\dot{x}\dot{z}+2z\dot{y}\dot{w}-2{\dot{y}}^2-3y\dot{z}\dot{w}+\dot{q}\dot{w}\\ \ddot{q}=\dot{x}\dot{w}.\end{array}l$(93)

The symmetry algebra is spanned by

(94) $\begin{array}{cc}{e}_1& =D_x,e_2=D_q,e_3=t{D}_p,e_4=D_p,e_5=t{D}_t,e_6=D_t,e_7=D_w,e_8=z{D}_p,\\ e_9& =w{D}_p,e_{10}=w{D}_t,e_{11}=z{D}_t,e_{12}=1/2tw{D}_p+t{D}_q,e_{13}=1/2w^2{D}_p+w{D}_q,\\ e_{14}& =q{D}_p+2x{D}_q+y{D}_x+z{D}_y,e_{15}=-3/2wz{D}_p+D_y,\\ e_{16}& =(wz-2y)D_t,e_{17}=1/2wz{D}_p+z{D}_q,e_{18}=(-1/2wz+y)D_p,\\ e_{19}& =(1/3w^3+2wz)D_p+w^2{D}_q+w{D}_x,\\ e_{20}& =3p{D}_p+5/2q{D}_q+1/2w{D}_w+2x{D}_x+3/2y{D}_y+z{D}_z,\\ e_{21}& =-1/4w(wz-2y)D_p+(-1/2wz+y)D_q,\\ e_{22}& =(-1/4z{w}^2+1/2wy+x)D_p-1/2wz{D}_q+D_z,\\ e_{23}& =(1/2z{w}^2-wy+2z^2+2x)D_p+wz{D}_q+z{D}_x,\\ e_{24}& =(1/12w^4+z{w}^2-3wy+2x)D_p+1/3w^3{D}_q+1/2w^2{D}_x+w{D}_y,\\ e_{25}& =(-1/6z{w}^3+1/60w^5-2z^{2}w-2q+wx+4yz)D_p+(-6x+1/12w^4-3/2z{w}^2+3wy)D_q\\ \hspace{0.17em}& +(-wz+1/6w^3)D_x+(-2z+1/2w^2)D_y+w{D}_z.\end{array}$(94)

The symmetry algebra g is a 25-dimensional indecomposable solvable Lie algebra. It has a 23-dimensional nilradical ${\mathrm{n}}_{23}$ spanned by

(95) $\begin{array}{cc}{e}_1,& e_2,e_3,e_4,e_6,e_7,e_8,e_9,e_{10},e_{11},e_{12},e_{13},e_{14},\\ e_{15},& e_{16},e_{17},e_{18},e_{19},e_{21},e_{22},e_{23},e_{24},e_{25}.\end{array}$(95)

and a two-dimensional abelian complement $\mathbb{R}{}^2$ spanned by

(96) $\begin{array}{cc}{e}_5,& e_{20}.\end{array}$(96)

We conclude that the symmetry Lie algebra is ${\mathrm{n}}_{23}\times {\mathrm{R}}^2$.

6.21: $[e_1,e_2]=e_3,[e_1,e_5]=e_6,[e_2,e_3]=e_4,[e_2,e_4]=e_5,[e_3,e_4]=e_6$:

$S=\left[\begin{array}{cccccc}1& z& -y+zw& x-yw+\frac{1}{2}z{w}^2& -q+xw+\frac{z{w}^3}{6}-\frac{y{w}^2}{2}& p\\ 0& 1& w& \frac{w^2}{2}& \frac{w^3}{3}& q\\ 0& 0& 1& w& \frac{w^2}{2}& x\\ 0& 0& 0& 1& w& y\\ 0& 0& 0& 0& 1& z\\ 0& 0& 0& 0& 0& 1\end{array}\right].$

Right-invariant vector fields: $D_z+q{D}_p,D_w+x{D}_q+y{D}_x+z{D}_y,D_y-x{D}_p,-(D_x+y{D}_p),D_q-z{D}_p,-2D_p$.

The geodesic equations are:

(97) $\begin{array}{c}\ddot{x}=\dot{y}\dot{w}\\ \ddot{y}=\dot{z}\dot{w}\\ \ddot{z}=0\\ \ddot{w}=0\\ \ddot{p}=z\dot{x}\dot{w}-y\dot{y}\dot{w}+x\dot{z}\dot{w}\\ \ddot{q}=\dot{x}\dot{w}.\end{array}$(97)

The symmetry algebra is spanned by

(98) $\begin{array}{cc}{e}_1& =t{D}_q,e_2=D_q,e_3=t{D}_p,e_4=D_p,e_5=t{D}_t,e_6=D_t,e_7=D_w,e_8=z{D}_q,\\ e_9& =w{D}_q,e_{10}=z{D}_p,e_{11}=w{D}_p,e_{12}=w{D}_t,e_{13}=z{D}_t,\\ e_{14}& =q{D}_p+D_z,e_{15}=x{D}_p+D_y,\\ e_{16}& =x{D}_q+y{D}_x+z{D}_y,e_{17}=1/2wz{D}_p+D_x,e_{18}=2p{D}_p+q{D}_q+x{D}_x+y{D}_y+z{D}_z,\\ e_{19}& =(wz-2y)D_t,e_{20}=(y-1/2wz)D_q,e_{21}=(y-1/2wz)D_p,\\ e_{22}& =3p{D}_p+3q{D}_q+w{D}_w+2x{D}_x+y{D}_y,\\ e_{23}& =(1/2z{w}^2-wy+2x)D_p+1/2w^2{D}_q+w{D}_x,\\ e_{24}& =(1/6z{w}^3-1/2y{w}^2+wx-2q)D_p+1/6w^3{D}_q+1/2w^2{D}_x+w{D}_y.\end{array}$(98)

The symmetry algebra g is a 24-dimensional indecomposable solvable Lie algebra. It has a 21-dimensional nilradical ${\mathrm{n}}_{21}$ spanned by

(99) $\begin{array}{cc}{e}_1,& e_2,e_3,e_4,e_6,e_7,e_8,e_9,e_{10},e_{11},e_{12},e_{13},e_{14},\\ e_{15},& e_{16},e_{17},e_{19},e_{20},e_{21},e_{23},e_{24}\end{array}$(99)

and a three-dimensional abelian complement $R{}^3$ spanned by

(100) $\begin{array}{cc}{e}_5,& e_{18},e_{22}.\end{array}$(100)

We conclude that the symmetry Lie algebra is ${\mathrm{n}}_{21}\times {\mathrm{R}}^3$.

6.22: $[e_1,e_2]=e_3,[e_1,e_3]=e_5,[e_1,e_5]=e_6,[e_2,e_3]=e_4,[e_2,e_4]=e_5,[e_3,e_4]=e_6$:

$S=\left[\begin{array}{cccccc}1& z& wz-y& x-yw+\frac{1}{2}z{w}^2-\frac{1}{3}{z}^2& xw-q-\frac{1}{2}y{w}^2+\frac{1}{3}zy+\frac{1}{6}z{w}^3-\frac{1}{3}{z}^{2}w& p\\ 0& 1& w& \frac{w^2}{2}-\frac{z}{3}& \frac{2}{3}y+\frac{1}{6}{w}^3-\frac{1}{3}zw& q\\ 0& 0& 1& w& \frac{w^2}{2}+\frac{z}{3}& x\\ 0& 0& 0& 1& w& y\\ 0& 0& 0& 0& 1& z\\ 0& 0& 0& 0& 0& 1\end{array}\right].$

Right-invariant vector fields: $D_z+\frac{1}{3}z{D}_x-\frac{1}{3}y{D}_q+q{D}_p,D_w+z{D}_y+y{D}_x+x{D}_q,D_y+\frac{2}{3}z{D}_q-x{D}_p,-D_x-$ $-y{D}_p,D_q-z{D}_p,-2D_p$.

The geodesic equations are:

(101) $\begin{array}{c}\ddot{x}=\dot{y}\dot{w}\\ \ddot{y}=\dot{z}\dot{w}\\ \ddot{z}=0\\ \ddot{w}=0\\ \ddot{p}=z\dot{x}\dot{w}-(1/3z+y)\dot{y}\dot{w}+(x-2/3z^2)\dot{z}\dot{w}-1/3y{\dot{z}}^2\\ \ddot{q}=-z\dot{z}\dot{w}+2\dot{x}\dot{w}.\end{array}$(101)

The symmetry algebra is spanned by

(102) $\begin{array}{cc}{e}_1& =t{D}_q,e_2=D_q,e_3=t{D}_p,e_4=t{D}_t,e_5=D_p,e_6=D_t,e_7=D_w,e_8=z{D}_q,\\ e_9& =w{D}_q,e_{10}=z{D}_p,e_{11}=w{D}_p,e_{12}=w{D}_t,e_{13}=z{D}_t,\\ e_{14}& =1/2wz{D}_p+D_x,e_{15}=(-1/6z^2-x)D_p+D_y,\\ e_{16}& (wz-2y)D_t,e_{17}=(y-1/2wz)D_q,e_{18}=(y-1/2wz)D_p,\\ e_{19}& =(1/2z{w}^2-wy+2x)D_p+w^2{D}_q+w{D}_x,\\ e_{20}& =(z{w}^3+(-3y-z)w^2+(-z^2+6x+4y)w-6q-8x)D_p+(2w^3+wz)D_q\\ \hspace{0.17em}& +(3w^2+z)D_x+6w{D}_y,\\ e_{21}& =(-5/12z{w}^3+(5/4y+5/12z)w^2+(5/12z^2-5/2x-5/3y)w+3q+3x)D_p\\ \hspace{0.17em}& +(-1/2wz-5/6w^3)D_q-5/4w^2{D}_x-5/2w{D}_y+D_z,\\ e_{22}& =(-1/6z{w}^3+(1/2y+1/6z)w^2+(1/6z^2-x-2/3y)w-1/18z^3+q+4/3x)D_p\\ \hspace{0.17em}& +(2x-1/3w^3)D_q+(y-1/2w^2)D_x+(-w+z)D_y.\end{array}$(102)

The symmetry algebra g is a 22-dimensional indecomposable solvable Lie algebra. It has a 21-dimensional nilradical ${\mathrm{n}}_{21}$ spanned by

(103) $\begin{array}{cc}{e}_1,& e_2,e_3,e_5,e_6,e_7,e_8,e_9,e_{10},e_{11},e_{12},e_{13},e_{14},\\ e_{15},& e_{16},e_{17},e_{18},e_{19},e_{20},e_{21}.\end{array}$(103)

and a 1-dimensional complement $R$ spanned by

(104) $\begin{array}{cc}{e}_4.& \end{array}$(104)

We conclude that the symmetry Lie algebra is${\mathrm{n}}_{21}\times \mathrm{R}$. We end this section with the following Theorem that summarizes the results obtained in this paper.

Theorem 3.1. The dimensions of the symmetry Lie algebras of the geodesic equations of the canonical connection on six-dimensional nilpotent Lie algebras are: 22, 24, 25, 26, 27, 28, 29, 30, 32, 34, 36, 40 and 63.

4. Conclusion

In this Section, we shall make some general qualitative remarks about the symmetry algebras that we have found. In the first place, if we look at the list of nilpotent Lie algebras itself, we see that, roughly speaking, the list proceeds from the most symmetric to the least symmetric algebras; in this context, by symmetric we do not necessarily mean that the Lie symmetry algebra is bigger. What we do mean is that the algebras that have larger centers tend to come first and correspondingly, the algebra contains fewer Lie brackets. Nonetheless, this decrease in symmetry is manifested in the symmetry algebras which too tend to decrease in dimension as one moves down the list. A related remark is that the last four algebras are filiform, so that the lower central series is of maximal length. In this regard algebra, 6.2 is an outlier in that it is the only other filiform in the list; on the other hand, it has a high degree of obvious symmetry in its own right. It is also significant that the symmetry algebras of each of the five filiforms have zero semi-simple part, which again is a manifestation of a lower degree of symmetry.

In general, we expect that the nilpotent Lie algebras and Lie groups will exhibit a higher degree of symmetry than their solvable counterparts. We believe that the explanation lies in the fact that all of the associated geodesic systems have at least two equations whose right-hand sides are zero and potentially more that could be made zero using an appropriate change of coordinates. In the solvable case, there is typically only one equation whose right-hand sides are zero. Roughly speaking, geodesic systems that have several such trivial equations manifest a greater degree of flatness than those systems that do not.

Acknowledgements

The authors thank Qatar Foundation and Virginia Commonwealth University in Qatar for funding this project.

Additional information

Funding

This work was supported by the Virginia Commonwealth University in Qatar/Qatar Foundation [FRG 2019].

Notes on contributors

Ryad Ghanam

Ryad Ghanam is a Professor of Mathematics at Virginia Commonwealth in Qatar. He works in the area of applied differential geometry and its applications, has published over 30 articles on mathematics. Gerard Thompson is a Professor of Mathematics at the University of Toledo. He works in the area of Differential Geometry interfacing Lie theory. He has published over 70 articles on mathematics research in a variety of journals.