As the proof of Lemma 2.1 in the above paper contains some doubtful statement that author is not sure if can provide other proof of, the author would like to apologize and provide some other proof of slightly weaker version of Lemma 2.2.
Lemma 0.7:
Let F be any field. Every matrix t from 
is a sum of at most four square-zero matrices.
Proof Let , and let’s define u as follows:
Clearly, , where , . Since , there exists such invertible triangular matrix u that that is equal to , the latter is equal to the sum , , , i.e. the sum of two square-zero matrices. The matrix is a negative direct sum of the matrices of the form , where is either a natural number or it is . Each of the matrices can be written as a sum of two square-zero matrices, so as a direct sum of them also can be written in such way.
Consequently, the final result should be stated as follows.
Theorem 0.8:
Suppose F is a field of characteristic different from 2. Every matrix from can be written as a sum of at most 12 square-zero matrices.
