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Picture below is from 70 of Jost's Riemannian Geometry and Geometric Analysis , How to get red line from the Lemma 2.4.1 ?

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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.

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  • $\begingroup$ Really a little trick, why I can't think of this ... :) $\endgroup$
    – Enhao Lan
    Nov 6, 2016 at 10:34

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