Matrices which Commute with their Transposes over a Field of Characteristic Two

Let M be a square matrix with entries in a field F of characteristic two. I show that a necessary and sufficient condition for M to be similar to a matrix over F which commutes with its transpose is that one of the following statements holds.

i) The minimal polynomial of M is not square-free.

ii) Some elementary divisor of M occurs with multiplicity greater than or equal to its degree of inseparability.