There is a theorem: Every finite-dimensional Clifford-Algebra representation $V$ is semisimple / completely reducible, which means that it's a direct sum of irreducible subrepresentations.
How this can be proven? Probably one has to construct to a given irreducible subrepresentation $U \subset V$ another subrepresentation $U' \subset V$, such that $U \oplus U' = V$. How?