On matrix sets invariant under conjugation and taking linear combinations of commuting elements

Subsets of a matrix algebra over a field that are invariant under conjugation and contain the linear span of each two of their commuting elements are described. They obviously include the subsets of diagonalizable and nilpotent matrices. In the paper, the case of an algebraically closed field is considered. The problem is easily reduced to description of subsets of diagonalizable matrices and subsets of nilpotent matrices with the given properties. So, among diagonalizable matrices, there are four of such subsets. As for the nilpotent case, it is proved that the subset should be defined by the condition that the sizes of all Jordan cells of the matrix belong to a certain number set. An explicit criterion is obtained in terms of this set.