When reading the paper by Kitaev (arXiv:0901.2686), it seems to me there is a certain kind of theorem roughly like this:
"Consider an algebra formed by $B_i$, $i=1,2...n$ with $[ B_i, B_j]_{s_{ij}} =2 \phi_{ij}$, where $\phi_{ij}=0$ for $i\neq j$, $\phi_{ii}\in \{\pm 1\}$, $s_{ij} \in \{\pm 1\}$ for $i\neq j$, $s_{ii}=+1$, and $[,]_{s_{ij}}$ means the commutation or anticommutation for $s_{ij}=-1$ or $s_{ij}=1$.
If there doesn't exist any product over some of the $B_i$s that commute with all $B_i$, then one can always find some products over $B_i$ to form $\gamma_{\mu}$, $\mu=1,2...n$, such that the $\gamma_{\mu}$ form a Clifford algebra
$\{\gamma_{\mu}, \gamma_{\nu}\}=2 \eta_{\mu \nu}$ where $ \eta_{\mu \nu} = diag(\underbrace{1,...,1}_{p},\underbrace{-1,...,-1}_q)$"
Could someone please comment on whether the theorem exists/where to find it or the direction for proving it? Thanks!