Finding an element in a Clifford algebra that satisfy some specific commutation and anti-commutation relation

Question

A Clifford algebra are the set of elements $\gamma^i$'s together with their multiplicative combinations and $\gamma^i$'s satisfy the anti-commutation relation $\{\gamma^i,\gamma^j\}=2g_{ij}$, where $g_{ij}$ denote the signature of the real space. My question is: if we are given some commutation relations to satisfy, for example in $\mathbb{R}^{1,10}$ (1 negative, 10 positive), given $$[\gamma,\gamma^i]=\{\gamma,\gamma^1\}=0\;\;(i\neq 1)$$ is there any general method to find $\gamma$ that satisfy the mentioned commutation and anti-commutation relations?