I am reading Riemannian Geometry and Geometric Analysis by Jurgen Jost. Let me mention the notations and what I have learned from the book.
In the book, the Clifford algebra $Cl(V)$ of a real vector field $V$ is defined to be the quotient of the tensor algebra $\bigoplus_{k\ge0} \bigotimes^k V$ by the two-sided ideal generated by elements of the form $v \otimes v + \langle v,v \rangle$.
The spin group $Spin(V)$ is defined as all elements of the form $$a = a_1...a_{2k} \text{ with } a_i \in V \text{, }\left\Vert a_i \right\Vert = 1 \text{ for } i = 1,...,2k $$
The chirality operator is $\Gamma = i^m e_1...e_n \in Cl^\mathbb{C}(V)$ where the upperscript $\mathbb{C}$ is the complexification, $m = \frac{n}{2} = \frac{dim_\mathbb{R}V}{2}$ and $e_i$ is an positive orthonormal basis (I am only interested in even dimensional case).
Since $\Gamma^2 = 1$, we may use $\Gamma$ to obtain a decomposition $Cl^\mathbb{C}(V)^\pm$ of $Cl^\mathbb{C}(V)$ into eigenspaces with eigenvalues $\pm 1$ under the multiplication of $\Gamma$.
Theorem 2.6.4 tells us that $Cl^\mathbb{C}(V)$ is isomorphic to the algebra of complex linear endomorphisms of the spinor space $S = \bigwedge W$. Under the isomorphism, $\Gamma$ equals $(-1)^k$ on $\bigwedge^k W$ so that we have the decomposition $$ S^\pm = {\bigwedge}^\pm W $$ where the +(-) sign on the right denotes elements of even (odd) degree.
I understand the material so far, but cannot see why this statement is true: Since $Spin(V)\subset Cl^+(V)$, $Spin(V)$ leaves the space $S^+$ and $S^-$ invariant.
I am thinking that for example, $e_1e_2\in Spin(V)$, but $\Gamma (e_1e_2) = (e_2e_1)\Gamma = i^m e_3...e_n $; Then why do we have the containment $Spin(V)\subset Cl^+(V)$? Do I misunderstand something? Even if we have the containment, why the spin group leaves $S^\pm \subset S$ invariant?
This is my first time seeing the Clifford algebra and representation, so I appreciate any relevant material for self-study.