  This work relates to three problems, the classification of maximal Abelian
subalgebras (MASAs) of the Lie algebra of square matrices, the classification
of 2-step solvable Frobenius Lie algebras and the Gerstenhaber's Theorem. Let M
and N be two commuting square matrices of order n with entries in an
algebraically closed field K. Then the associative commutative K-algebra, they
generate, is of dimension at most n. This result was proved by Murray
Gerstenhaber in 1961. The analog of this property for three commuting matrices
is still an open problem, its version for a higher number of commuting matrices
is not true in general. In the present paper, we give a sufficient condition
for this property to be satisfied, for any number of commuting matrices and
arbitrary field K. Such a result is derived from a discussion on the structure
of 2-step solvable Frobenius Lie algebras and a complete characterization of
their associated left symmetric algebra structure. We discuss the
classification of 2-step solvable Frobenius Lie algebras and show that it is
equivalent to that of n-dimensional MASAs of the Lie algebra of square
matrices, admitting an open orbit for the contragradient action associated to
the multiplication of matrices and vectors. Numerous examples are discussed in
any dimension and a complete classification list is supplied in low dimensions.
Furthermore, in any finite dimension, we give a full classification of all
2-step solvable Frobenius Lie algebras corresponding to nonderogatory matrices.
