I have not dealt with the gamma matrices extensively so I am having a bit of trouble here.
Basically I want to show that the spin operator defined by $$ \mathbf{\hat{S}} = \frac{1}{2}\gamma^5 \gamma^0 \boldsymbol{\gamma}$$
saitisfies the commutation relation $ [H,\mathbf{S}] = \gamma^0 \boldsymbol{\gamma} \times \nabla$ with the Hamiltonian: $$ H = \gamma^0(-i\boldsymbol{\gamma}\cdot\nabla + m) .$$
My work so far:
$$ [H,\mathbf{S} ]\color{blue}{\psi} = \\ H\mathbf{S}\color{blue}{\psi} - \mathbf{S} H\color{blue}{\psi} = \\ \gamma^0(-i\boldsymbol{\gamma}\cdot\nabla + m)*\frac{1}{2}\gamma^5 \gamma^0 \boldsymbol{\gamma}\color{blue}{\psi} - \frac{1}{2}\gamma^5 \gamma^0 \boldsymbol{\gamma}* \gamma^0(-i\boldsymbol{\gamma}\cdot\nabla + m)\color{blue}{\psi} = \\ -i\gamma^0\boldsymbol{\gamma}\cdot {\nabla \left ( \frac{1}{2}\gamma^5 \gamma^0 \boldsymbol{\gamma}\color{blue}{\psi} \right )} + {\frac{m}{2}\gamma^5 \gamma^0 \boldsymbol{\gamma}}\color{blue}{\psi} + \frac{i}{2}\gamma^5 \gamma^0 \boldsymbol{\gamma} \gamma^0 \boldsymbol{\gamma}\cdot\nabla\color{blue}{\psi} - \frac{m}{2}\gamma^5 \gamma^0 \boldsymbol{\gamma}\color{blue}{\psi} = \\ -i\gamma^0\boldsymbol{\gamma}\cdot {\nabla \left ( \frac{1}{2}\gamma^5 \gamma^0 \boldsymbol{\gamma}\color{blue}{\psi} \right )} + \frac{i}{2}\gamma^5 \gamma^0 \boldsymbol{\gamma} \gamma^0 \boldsymbol{\gamma}\cdot\nabla\color{blue}{\psi} $$
switching to index notation now $[H, S^i]$ :
$$ \frac{-i}{2}\gamma^0 \gamma^5\gamma^0 \gamma^i \gamma^k\partial^k + \frac{i}{2}\gamma^5\gamma^0\gamma^i\gamma^0 \gamma^j\partial^j, $$ rearranging: $$ -i\gamma^5\gamma^i\gamma^j\partial^j $$
Now, the answer is $ \gamma^0 \boldsymbol{\gamma} \times\nabla$, and to get the $ \times $ in there I need a Levi-Civita symbol. Which I guess comes from the definition of $$ \gamma^5 = \frac{i}{4!}\epsilon_{\mu\nu\alpha\beta}\gamma^{\mu}\gamma^{\nu}\gamma^{\alpha}\gamma^{\beta}, $$ from which I would have $$[H, S^i] = \frac{1}{4!}\epsilon_{\mu\nu\alpha\beta}\gamma^{\mu}\gamma^{\nu}\gamma^{\alpha}\gamma^{\beta} \gamma^i \gamma^j \partial^j $$ where the greek letters run from $0$ to $4$ whereas the latin ones only from $1$ to $3$.
How do I proceed?