Spin Operator in terms of Gamma-5 The QM spin operator can be expressed in terms of gamma matrices and I am trying to do an exercise where I prove an identity which uses $\gamma^5$ and ${\mathbf{\alpha}}$:
$$\mathbf{S}=\frac{1}{2}\gamma^5\mathbf{\alpha}$$
In my first attempt I did this directly in the Dirac representation but the exercise states that I cannot do this, can anyone advise?  Is there some identity or trick which would enable me to do this?
To clarify, $\alpha$ is the following matrix where the non-zero elements are the Pauli matrices:
$
   \alpha^i=
  \left[ {\begin{array}{cc}
   0 & {\sigma^i} \\
   {\sigma^i} & 0 \\
  \end{array} } \right]
$
$\textbf{S}=\frac{1}{2}\Sigma$
where
$
  \Sigma= 
  \left[ {\begin{array}{cc}
   {\sigma^i} & 0 \\
   0 & {\sigma^i} \\
  \end{array} } \right]=-i\alpha_{1}\alpha_{2}\alpha_{3}\mathbf{\alpha}
$
 A: I'm following the conventions of Wikipedia with the following definitions
$$
\Sigma^{\mu\nu} =   \frac{i}{4} [ \gamma^\mu , \gamma^\nu ] , \qquad S^i = \frac{1}{2} \epsilon^{ijk} \Sigma^{jk}, \qquad \alpha^i = \gamma^0 \gamma^i , \qquad \gamma^5 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3 .
$$
where
$$
\{ \gamma^\mu , \gamma^\nu \} = 2 \eta^{\mu\nu} , \qquad \eta^{\mu\nu} = \text{diag}(1,-1,-1,-1).
$$
Having said this, we now note
$$
S^i =   \frac{i}{4} \epsilon^{ijk}\gamma^j\gamma^k
$$
Explicitly,
$$
S^1 =  \frac{i}{2} \gamma^2 \gamma^3 , \qquad S^2 = \frac{i}{2} \gamma^3 \gamma^1, \qquad S^3 =  \frac{i}{2} \gamma^1 \gamma^2
$$
Then,
$$
\frac{1}{2} \gamma^5  \alpha^1 = \frac{1}{2}  i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \gamma^0 \gamma^1 =  \frac{i}{2}   \gamma^2 \gamma^3  = S^1 , \\
\frac{1}{2} \gamma^5  \alpha^2 = \frac{1}{2}  i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \gamma^0 \gamma^2 =  - \frac{i}{2}   \gamma^1 \gamma^3  = S^2 , \\
\frac{1}{2} \gamma^5 \alpha^3 = \frac{1}{2}  i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \gamma^0 \gamma^3 =  - \frac{i}{2}   \gamma^1 \gamma^2  = S^3 , \\
$$
Thus,
$$
S^i = \frac{1}{2} \gamma^5 \alpha^i.
$$
