Exercise 5, § 14

Let K be a commutative field, V a vector space of finite dimension $n$ over K, and W a subspace of V distinct from $\{0\}$ and V. Denote by $\mathfrak{a}$ the subset of the algebra A = End$_K(V)$ consisting of the endomorphisms $u$ such that Im $u\subset W\subset$ Ker $u$ and by B the subalgebra $K +\mathfrak{a}$ of A. Prove that B is a maximal commutative subalgebra of A, is not semisimple, and has degree possibly $> n$ over K.