I want to show that $\mathfrak{so}(4)\cong \mathfrak{so}(3)\oplus \mathfrak{so}(3)$. I know that as lie groups $SO(4)\cong (SU(2)\times SU(2))/\mathbb{Z}_2$ and that as $SU(2)/\mathbb{Z}_2 \cong SO(3)$.
My idea to do this was to show that $SO(4)\cong SU(2)\mathbb{Z}_2\times SU(2)/\mathbb{Z}_2$ and then the result should follow. But the map from $(SU(2)\times SU(2))/\mathbb{Z}_2$ to $SU(2)\mathbb{Z}_2\times SU(2)/\mathbb{Z}_2$ is only surjective not injective. Hence the map from $SO(4)$ to $SU(2)\mathbb{Z}_2\times SU(2)/\mathbb{Z}_2$ is not an isomorphism.
Is this the wrong approach or have I just made a mistake?
Lie algebra structure only uniquely determines the connected component of the identity a lie group. If one shows that the homomorphism restricts to an isomorphism on the connected component of the identity then this would do.
But as suggested it is perhaps easier to d this all on the level of lie algebras.