Real independence of Clifford-generated vectors

For hermitian generators satisfying $\{{\gamma }^I,{\gamma }^J{\}}_{+}=2{\delta }^{IJ}$ and any nonzero vector $v$, the vectors ${\gamma }^{I}v$ are linearly independent over the real numbers. Indeed, $A={\sum }_I{a}_I{\gamma }^I$ with real $a_I$ obeys $A^2=({\sum }_I{a}_I^2)1$, so $\Vert Av{\Vert }^2=({\sum }_I{a}_I^2)\Vert v{\Vert }^2$. Complex coefficients need a different argument.