The char. polynomial of a nilpotent 4x4 matrix is , the question now is: what's its minimal polynomial ?

For example, if there's at least one Jordan Block (JB) of order 2, so you have two possibilities for the JNF of a matrix like this:

either --- one 2x2 JB + two 1x1 JB's , or

--- two 2x2 JB's.

Well, now check the other possibilities for the minimal polynomial.

Tonio