Does a Lie group product such as $SU(2)\times SU(2)$ translate to a sum of Lie algebras, and vice-versa? For example, consider the Lie algebra $su(2)+isu(2)$. When exponentiated, we get
$$ e^{su(2)+isu(2)} = e^{su(2)}e^{isu(2)} = SU(2)\times SU(2) $$
where the $i$ is added, I guess, to guaranteed commutativity of the two terms $su(2)$ and therefore split them out of the exponentiation?