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Suppose we write the gamma matrices in this following representation:

\begin{align*} \gamma^{0}=\begin{pmatrix} \,\,0 & \mathbb{1}_{2}\,\,\\ \,\,\mathbb{1}_{2} & 0\,\, \end{pmatrix},\qquad \gamma^{i}=\begin{pmatrix} \,\,0 &\sigma^{i}\,\,\\ \,\,-\sigma^{i} & 0\,\, \end{pmatrix} \end{align*} where $i=1,2,3$. Consider $2k+2$ spacetime dimension, we can then group the gamma matrices into $k+1$ anticommuting 'raising' and 'lowering' operators, defined by \begin{align*} \Gamma^{0\pm} = \frac{i}{2}\left(\pm\gamma^{0}+\gamma^{1}\right),\qquad \Gamma^{a\pm}=\frac{i}{2}\left(\gamma^{2a}\pm i\gamma^{2a+1}\right) \end{align*} with $a=1,2,...,k$. The anticommutation relations satisfy \begin{align*} \left\{\Gamma^{a\pm},\Gamma^{b\pm}\right\}=0,\qquad \left\{\Gamma^{a\pm},\Gamma^{b\mp}\right\} =\delta^{ab} \end{align*}

Then how can I construct the gamma matrices in $6$ and $8$ dimensions using this information?

I expect $8\times 8$ matrices for $6$ dimension and $16\times 16$ matrices for $8$ dimension. The thing that keeps bugging me is how we supposed to construct the matrices in higher dimensions with these operators and their anticommutation relations? This was one of the homework questions but I'm just curious how this thing works.

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