Let $V$ be a vector space with dimension $n$ and $q$ a quadratic form on $V$. How many involutive automorphisms are there in $\mathcal{C} \ell (V,q)$?
2026-03-27 14:39:50.1774622390
Number of Involutive Automorphisms on a Clifford Algebra
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In general, infinitely many.
Take for example $\Bbb H$, which is a Clifford algebra for a suitable choice of metric on $\Bbb R^2$. Pick any unit length quaternion $u$ with real part $0$. Then $(u\cos(\pi/2)-\sin(\pi/2))^2=1$.
There are as many choices for $u$ as there are points on the unit sphere in $3$-space.
Each of these will describe an involution given by $x\mapsto uxu^{-1}$