In Deligne's 1972 article on the Weil conjecture for K3 surfaces, he essentially constructed an inclusion of Shimura data $(GSpin(V),X)\subset(GSp(W),H(W))$, where $V$ is a vector space over $\mathbb{Q}$ of dimension $n+2$ endowed with a quadratic form $q$ of signature $(n,2)$, $C(V)$ the Clifford algebra associated to $(V,q)$, $C^+(V)$ the even part of $C(V)$, $GSpin(V)$ the reductive $\mathbb{Q}$-group of invertible elements in $C^+(V)$ that preserve $V$ under conjugation in $C(V)$. The Hermitian symmetric domain $X$ can be identified as the space of $q$-isotropic negative planes in $V_\mathbb{R}$, as is described in Kudla's article "algebraic cycles in Shimura varieties of orthogonal type". The inclusion of Shimura data mentioned above is given by the representation $(W,\rho_W)$ by left translation of $GSpin(V)\subset C^+(V)$ on $W=C^+(V)$, which respects a canonically defined symplectic pairing up to scalar. Finally $H(W)$ is the Siegel double space associated to the above symplectic structure.

Note that for any $x\in X$ (or in $H(W)$), $(W,\rho\circ x)$ is a Hodge structure of type $(-1,0),(0,-1)$, hence $W\otimes W$ underlies a Hodge structure of type $(-2,0),(-1,-1),(0,-2)$. On the other hand, the canonical representation $(V,\rho_V)$ of $GSpin(V)$ gives Hodge structures $(V,\rho_V\circ x)$ of type $(-2,0),(-1,-1),(0,-2)$.

My question : is there a natural embedding of $V$ into $W\otimes W$ as a subrepresentation of $GSpin(V)$? if there is, then I can naturally understand $V$ as a Hodge substructure of $W\otimes W$. The Hodge types already coincile, but I don't know if it follows from some simple universal construction.