On Wikipedia and often in physics books it is stated, that the Lorenz group's finite dimensional non-unitary representations are completely reducible. We heavily use this in physics, however I do not know of a direct proof or reference for this. Could anyone help in providing such? It is clear to me that the Lorentz algebra $so(3,1)$ is semisimple, thus completely reducible. However, I don't see how we go to the group level, cause the exponential map is not surjective, since $SO(3,1)$ is non-compact and disconnected.
In particular, this is of interest because fields transform in irreps of $SO(3,1)$, so complete reducibility would be useful to have.