In many books on particle physics, it is first shown that the Lie algebra of the Lorentz group is isomorphic to
$$ su(2) \oplus su(2) , $$
then by this fact, it is implicitly assumed that for each representation of $su(2)\oplus su(2)$, one can construct a representation of the Lorentz group.
But it is well known that the group $SO(3)$ has a Lie algebra isomorphic to $su(2)$, but half of the latter's representations do not lead to a representation of the former.
The question is then, does every representation of $su(2)\oplus su(2)$ lead to a representation of the Lorentz group? even though it cannot be unitary?
In particular, does the representation labeled as $(1/2, 0)$ lead to a representation of the group?
