 Dimension bound from paired Clifford involutions

For $2n$ hermitian Euclidean Clifford generators, the commuting hermitian involutions $J_j=i{\gamma }^{2j-1}{\gamma }^{2j}$ admit a common eigenvector. Each odd-indexed generator flips one joint eigenvalue. The $2^n$ resulting vectors are nonzero and mutually orthogonal. Thus any nonzero invariant complex representation has dimension at least $2^n$.