A minimal example of the Clifford algebra is the $\mathbb{C}$-algebra (unital, associative) generated by $x,y$ quotient over the relations \begin{eqnarray} x^2&=&1,\tag{1}\\ y^2&=&1,\tag{2}\\ xy&=&-yx.\tag{3} \end{eqnarray} It's easy to see that this is a 4-dimensional algebra with basis $1,x,y,xy$. Notice, it is important that Eq.(3) alone guarantees that $[x^2,y]=[y^2,x]=0$--for example, if we replace Eq.(3) by $xy=\theta yx$ with constant $\theta\neq \pm 1$, we would have an inconsistency $x=xy^2=\theta^2 y^2x=\theta^2x$ leading to $x=0$.
Here is a natural generalization of the algebra above. Let $m$ be a positive integer and let $W$ be the $\mathbb{C}$-algebra generated by $\{x_i,y_i\}_{1\leq i\leq m}$ quotient over the relations \begin{eqnarray} \sum_{1\leq i,j\leq m}x_i s_{ij} x_j&=&1,\tag{1'}\\ \sum_{1\leq i,j\leq m}y_i s_{ij} y_j&=&1,\tag{2'}\\ \sum_{1\leq k,l\leq m} S_{lk}^{ji} y_k x_l&=&x_iy_j,\tag{3'} \end{eqnarray} where $s_{ij}$ is a constant $m\times m$ matrix and $S_{lk}^{ji}$ is a constant 4-index-tensor. The tensors $S,s$ are required to satisfy $\sum_k s_{jk} S^{ik}_{lp}=\sum_kS^{ki}_{pj}s_{kl}$. With this condition, it is straightforward to show that Eq.(3') alone guarantees that $[y_i, x.s.x]=[x_i,y.s.y]=0$ (I use "." to denote tensor index contraction when there's no confusion), reminiscent of the consistency of the Clifford algebra. I also require $\sum_{k,l}S^{ij}_{kl}S^{kl}_{pq}=\delta_{ip}\delta_{lq}$, so that there is an automorphism exchanging $x_i\leftrightarrow y_i$. Notice that setting $m=1$, $S=-1$ and $s=1$ gives back the Clifford algebra.
Questions:
(1). Let $U$ be the algebra generated by $\{x_i\}^m_{i=1}$ quotient over the relation $x.s.x=1$. Then, is $U$ isomorphic to the subalgebra of $W$ generated by its elements $\{x_i\}^m_{i=1}$? The answer is not obvious since it is possible that Eqs.(2',3') may imply additional relations on $\{x_i\}^m_{i=1}$. This question is important for working out a basis for $W$, since if the answer is yes, then we would have a vector space isomorphism $W\cong U\otimes U$, as for any monomial in $x_i,y_j$, one can always use Eq.(3') to move the $x_i$s to the left and the $y_j$s to the right.
(2). Under what further conditions on $S,s$ does $W$ have a finite dimensional representation?