In T. Friedrich, "Dirac Operators in Riemannian Geometry", the author gives the following exercice (p34) :
Prove that the group ${\rm Spin}_{\mathbb C}(4)$ is isomorphic to the following subgroup $H$ of $U(2)\times U(2)$: $$H = \{(A, B) \in U(2)\times U(2) : \det (A) = \det (B)\}.$$
This surprised me because in all other textbooks we have ${\rm Spin}_{\mathbb C}(4)\simeq SL_2(\mathbb C)\times SL_2(\mathbb C).$
How can it be that $U(2)\subset SL_2(\mathbb C)$ and $SL_2(\mathbb C)\times SL_2(\mathbb C)\subset U(2)\times U(2)$ ? Did I miss something or is that exercice impossible ?
You might be confusing the compact group ${\rm Spin}_{\mathbb C}(4)$ and the non-compact complex group ${\rm Spin}(4,\mathbb{C})$.