<s>[INST] How many distinct possible forms for its Jordan canonical matrix are there? 4x4 non-diagonalizable matrix with two unique eigenvalues I know the sum of  $A_m$ equals $4$ as $\dim(A) = 4$ and sum of $G_m$ can't equal $4$ as $A$ is non-diagonalizable. 
After I write down all the cases, what should I do? 
 [/INST] After you write down the cases, count how many you have. I count 24.  Following is my work where the first entry in an ordered pair refers to the first eigenvalue, and the second to the second distinct eigenvalue.  Note that without a convention regarding Jordan canonical form, blocks can be permuted.
\begin{array}{cccrl}
\text{algebraic} & \text{geometric} & \text{Jordan}\\
\text{multiplicity} & \text{multiplicity} & \text{block sizes} & \text{permutations} \\ \hline
(1, 3) & (1, 1) & (1, 3)                & 2!    = 2 \\
(1, 3) & (1, 2) & (1, 1 \text{ and } 2) & 3!    = 6 \\
(2, 2) & (1, 1) & (2, 2)                & 2!    = 2 \\
(2, 2) & (1, 2) & (2, 1 \text{ and } 1) & 3!/2! = 3 \\
(2, 2) & (2, 1) & (1 \text{ and } 1, 2) & 3!/2! = 3 \\
(3, 1) & (1, 1) & (3, 1)                & 2!    = 2 \\
(3, 1) & (2, 1) & (1 \text{ and } 2, 1) & 3!    = 6 \\ \hline
       &        &                       &        24 & \text{total}
\end{array}
</s>