Universal Clifford algebra of a quadratic space

Let $k$ be a field with $\mathrm{char}(k)\mathrm{\ne }2$, let $V$ be a finite-dimensional $k$-vector space, and let $B:V\times V\to k$ be a symmetric bilinear form, not necessarily nondegenerate. Put $Q(v)=B(v,v)$. Write $T_k(V)$ for the tensor algebra and $j:V\to T_k(V)$ for its degree-one inclusion (Tensor algebra of a vector space). For each $v\in V$, set $r_v=j(v)j(v)-Q(v)1$ and let $J_Q$ be the intersection of all two-sided ideals of $T_k(V)$ containing every $r_v$ (Left, right and two-sided ideals). This family is nonempty because it contains $T_k(V)$, and its intersection is a two-sided ideal; thus $J_Q$ is the smallest such ideal.

Define the universal Clifford algebra with the plus-$Q$ convention by

${\mathrm{C}\mathrm{l}}^{+}(V,Q):=T_k(V)\mathrm{/}{J}_Q\mathrm{.}$

The quotient operations are induced from $T_k(V)$; the two-sided ideal condition makes multiplication of cosets well defined. Let $q_Q:T_k(V)\to {\mathrm{C}\mathrm{l}}^{+}(V,Q)$ be the quotient map, put $i_Q=q_Q\circ j$, and write $\bar{v}=i_Q(v)$. The defining relation is ${\bar{v}}^2=Q(v)1$. In this pair, $\mathrm{C}\mathrm{l}(V,Q)$ abbreviates ${\mathrm{C}\mathrm{l}}^{+}(V,Q)$ after this sign convention has been fixed. No nondegeneracy assumption is imposed.