How does the following representation of $\mathbb{C}l(3)$ decompose into irreducibles?

Question

Consider the three dimensional complex Clifford algebra $\mathbb{C}l(3)$ and the following representation $$S=\text{Span}\{|0\rangle,c^\dagger|0\rangle,\gamma^3|0\rangle,\gamma^3c^\dagger|0\rangle\},$$ where $c=\partial/\partial c^\dagger$ (with $c$ and $\gamma^3$ anticommuting variables) and $(\gamma^3)^2=1$. This furnishes a four dimensional representation by $$\gamma^1=c^\dagger+c$$ and $$\gamma^2=-i(c^\dagger-c).$$ But $\mathbb{C}l(3)$ has a unique irreducible representation, and it happens to be two dimensional. How can I express $S$ as a direct sum of two copies of this irrep?

Answer 1

I managed to play around a bit and found the decomposition! It is $$S=\text{Span}\{|0\rangle+\gamma^3|0\rangle,c^\dagger|0\rangle-\gamma^3c^\dagger|0\rangle\}\oplus\text{Span}\{|0\rangle-\gamma^3|0\rangle,c^\dagger|0\rangle+\gamma^3c^\dagger|0\rangle\}$$