The complex spin groups $Spin^C(n)$ appear in the fibration

$Spin(n)\rightarrow Spin^C(n)\rightarrow\ S^1$

which must split since $BSpin(n)$ is 3-connected to give a homotopy equivalence

$Spin^C(n)\simeq Spin(n)\times S^1$

This is, however, only an equivalence of spaces, and not of topological groups.

In low dimensions there are the 'accidental' equivalences of lie groups $Spin(3)\cong SU(2)$, $Spin(4)\cong SU(2)\times SU(2)$, $Spin(5)\cong Sp(2)$ and $Spin(6)\cong SU(4)$.

My question is if these lead to 'accidental' isomorphisms of groups $Spin^C(3)\cong U(2)$, $Spin^C(4)\cong U(2)\times SU(2)$ and $Spin(6)\cong SU(4)$?

The first of these isomorphism appears in the literature but I cannot find references for the other two above conjectures. They are certainly true homotopically, but are they true as topological groups?

The complex spin groups $Spin^C(n)$ appear in the fibration

$Spin(n)\rightarrow Spin^C(n)\rightarrow\ S^1$

which must split since $BSpin(n)$ is 3-connected to give a homotopy equivalence

$Spin^C(n)\simeq Spin(n)\times S^1$

This is, however, only an equivalence of spaces, and not of topological groups.

In low dimensions there are the 'accidental' equivalences of lie groups $Spin(3)\cong SU(2)$, $Spin(4)\cong SU(2)\times SU(2)$, $Spin(5)\cong Sp(2)$ and $Spin(6)\cong SU(4)$.

My question is if these lead to 'accidental' isomorphisms of groups $Spin^C(3)\cong SU(2)$, $Spin^C(4)\cong U(2)\times SU(2)$ and $Spin(6)\cong SU(4)$?

The first of these isomorphism appears in the literature but I cannot find references for the other two above conjectures. They are certainly true homotopically, but are they true as topological groups?