In both the Weyl and Dirac basis, the Dirac matrices satisfy
$
\hat \gamma_k \hat \gamma_0 = \hat \gamma_5 \hat\sigma_k
$
where $\hat \sigma_k = \begin{bmatrix}\sigma_k & 0 \\ 0 & \sigma_k \end{bmatrix}$, whereas in the Hestenes spacetime split, we define
$
\sigma_k \equiv \gamma_k \gamma_0.
$
This may lead to confusion.
## Dirac basis
![[Dirac Basis]]
$\hat \gamma_k \hat \gamma_0 = \begin{bmatrix}
0 & -\sigma_k \\
\sigma_k & 0
\end{bmatrix} \begin{bmatrix}
I_2 & 0 \\
0 & -I_2
\end{bmatrix} = \begin{bmatrix} 0 & \sigma_k \\ \sigma_k & 0\end{bmatrix} = \hat \gamma_5 \hat \sigma_k$
## Weyl basis
![[Weyl Basis]]
$\hat \gamma_k \hat \gamma_0 = \begin{bmatrix}
0 & -\sigma_k \\
\sigma_k & 0
\end{bmatrix} \begin{bmatrix}
0 & I_2 \\
I_2 & 0
\end{bmatrix} = \begin{bmatrix}
-\sigma_k & 0 \\
0 & \sigma_k
\end{bmatrix} = \hat \gamma_5\hat\sigma_k$