Answer
See below
Work Step by Step
There are 3 possible Jordan canonical forms:
$J_1=\begin{bmatrix}
-1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0\\
0 & 0 & -1 & 0\\
0 & 0 & 0 & 2
\end{bmatrix}$
There are 4 linearly independent eigenvectors of a matrix with this Jordan canonical form, and the maximum length of a cycle is $1$
$J_2=\begin{bmatrix}
-1 & 1 & 0 & 0 \\
0 & -1 & 1 & 0\\
0 & 0 & -1 & 0\\
0 & 0 & 0 & 2
\end{bmatrix}$
There are 3 linearly independent eigenvectors of a matrix with this Jordan canonical form, and the maximum length of a cycle is $2$
$J_3=\begin{bmatrix}
-1 & 1 & 0 & 0 \\
0 & -1 & 0 & 0\\
0 & 0 & -1 & 0\\
0 & 0 & 0 & 2
\end{bmatrix}$
There are 2 linearly independent eigenvectors of a matrix with this Jordan canonical form, and the maximum length of a cycle is $3$
