For an $n$-dimensional space $V$ with a positive metric $g$, we can construct the Clifford algebra $Cl(V)$ and its representation space $S$, i.e. $$c(V):S\to S,~\forall v\in V.$$

Question: Under what condition, we can find an another Clifford structure $\tilde{Cl}(V)$ acting on $S$, such that these two Clifford structures skew commute, i.e. $$c(v)\tilde c(w)+\tilde c(w)c(v)=0,~~\forall v,w\in V.$$


I'm not sure what you're asking. You might be asking:

Potential interpretation 1: Can we find a second embedding $\tilde c: V \to \mathrm{Cliff}(V)$ so that the images of $c$ and $\tilde c$ commute with each other?

Answer 1: No (unless $\dim V = 0$). The graded center of $\mathrm{Cliff}(V)$ is just the scalars, and $c(V)$ generates, so anything commuting with all of $c(V)$ is scalar.

But you might be asking:

Potential interpretation 2: Can we find a second Clifford algebra $\widetilde{\mathrm{Cliff}}(V)$, for the same $V$, that acts on $S$ commuting with the $\mathrm{Cliff}(V)$ action?

Missing information: What, exactly, is $S$?

My best guess is that $S$ is the irreducible $\mathrm{Cliff}(V)$ module. If so, the commutant of $\mathrm{Cliff}(V)$ in $\mathrm{End}(S)$ is the simple superalgebra in the Morita class of $\mathrm{Cliff}(V)$, and so:

Answer 2: Yes if $\dim V < 4$, no otherwise, as these are the only times when $\mathrm{Cliff}(V)$ is already simple. (For complex Clifford algebras, you need $\dim V <2$.)

Remark: By Bott periodicity, the simple superalgebra in the Morita class of $\mathrm{Cliff}(V)$ is $$ \begin{array}{c|c} \dim V& \mathrm{End}_{\mathrm{Cliff}(V)}(S) \\ \hline 8k & \mathbb R = \mathrm{Cliff}(0) \\ 1+8k & \mathrm{Cliff}(1) \\ 2+8k & \mathrm{Cliff}(2) \\ 3+8k & \mathrm{Cliff}(3) \\ 4+8k & \mathbb H \\ 5+8k & \mathrm{Cliff}(-3) \\ 6+8k & \mathrm{Cliff}(-2) \\ 7+8k & \mathrm{Cliff}(-1) \end{array} $$ where $\mathbb H$ is the purely-even quaternions and $\mathrm{Cliff}(n) = \mathrm{Cliff}(\mathbb R^n)$ with positive-definite metric and $\mathrm{Cliff}(-n) = \mathrm{Cliff}(\mathbb R^n)$ with negative-definite metric.


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