Weak polynomial identities for a vector space with a symmetric bilinear form

Let $V_k$ be a $k$-dimensional vector space with a non-degenerate symmetric bilinear form over a field $K$ of characteristic 0 and let $C_k$ be the Clifford algebra on $V_k$. We study the weak polynomial identities of the pair $(C_k,V_k)$. We establish that all they follow from $[x_1^2,x_2]=0$ when $k=\infty$ and from $[x_1^2,x_2]=0$ and $S_{k+1}(x_1,\ldots,x_{k+1})=0$ when $k<\infty$. We also prove that the weak identity $[x_1^2,x_2]=0$ satisfies the Specht property. As a consequence we obtain a new proof of the theorem of Razmyslov that the weak Lie polynomial identities of the pair $(M_2(K),sl_2(K))$ follow from $[x_1^2,x_2]=0$.