On sesquilinear forms over fields with involution

Let k be a field of characteristic ≠2 with an involution σ. A matrix A is split if there is a change of variables Q such that (Q^σ)^TAQ consists of two complementary diagonal blocks. We classify all matrices that do not split. As a consequence we obtain a new proof for the following result. Given a square matrix A there is a matrix S such that (S^σ)^TAS=A^T and S^σS=I.