Picture below is from 70 of Jost's Riemannian Geometry and Geometric Analysis , How to get red line from the Lemma 2.4.1 ?
2026-03-26 06:25:38.1774506338
Center of Spin(V)
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By definition, $\text{Spin}(V)$ is subset of even part of the Clifford algebra. So if $n$ is even $a \in Cl^0(V)$ and is a real number, if $n$ is odd $a \in (Cl^0(V) \oplus Cl^n(V)) \cap \text{Spin}(V) \subset Cl^0(V)$ and is again a real number.