Given a linear transformation $L:V→V$ where $V$ has basis $\{v_1, v_2, v_3\}$ with $L(v_i) = v_i$ for all $i < 3$ and $L(v_3) = v_1 + v_2$, decompose $V$ as a direct sum of $T$-cyclic subspaces as in the cyclic decomposition theorem.
The proof for the theorem was very long and complicated so I do not know how to actually use it in cases like this. Is there an algorithm for doing this? Or does the proof only show existence and does not actually tell us how to construct these cyclic subspaces?