It is well known that SU2 is locally isomorphic to SO3. U is just the Kronecker product. It is well known that SU2 is locally isomorphic to SO3.
Lie groups can only be locally isomorphic if their Lie algebras are isomorphic.
In all these examples the Lie algebra will be semisimple and so the Lie algebras in question can only be isomorphic if their root systems are identical. For details see any text on Lie groups. Before you buy.