Showing that a binary linear code $C$ is self-dual
$G^*$ is also a parity check matrix of the code $C^*$ since $G^*G^{*T}=0$ and $\dim\ker\left(G^{*}\right)=\text{rank}\left(G^*\right)=4$. 
 Since a parity check matrix of $C^*$ is a generator matrix of $C^{*\perp}$, we get that $C^*$ is a self dual code.