Maximal commutative subalgebras of matrix algebras

Maximal commutative subalgebras of the algebra of n by n matrices over a field k very rarely have dimension smaller than n. There is a (B, N)-construction which yields subalgebras of this kind. The Courter's algebra that is of this kind was shown a (B, N)-construction where B is the Schur algebra of size 4 and N = k(4). That is, the Courter's algebra is isomorphic to B x (k(4))(2), the idealization of (k(4))(2). It was questioned how many isomorphism classes can be produced by varying the finitely generated faithful B-module N. In this paper, we will show that the set of all algebras B x N-2 fall into a single isomorphism class, where B is the Schur algebra of size 4 and N a finitely generated faithful B-module.