We have been asked to show that there is a relation between the $(1/2,0)\oplus(0,1/2)$ (Dirac rep $R_D$) and the $(1/2,1/2)$ (vector rep $R_V$) representations of the Lorentz group:
$$ R_D(M)\gamma_\mu R_D(M)^{-1}=R_V(M)^\nu_\mu \gamma _\nu ,$$ where $M$ is just some element in the Lorentz group.
The only hint we are given is that the Lie algebras of the Lorentz group and $SL(2,\mathbb{C})$ are isomorphic.