Spin Group - Accidental Isomorphisms

In low dimensions, there are isomorphisms among the classical Lie groups called accidental isomorphisms. For instance, there are isomorphisms between low dimensional spin groups and certain classical Lie groups, owing to low dimensional isomorphisms between the root systems (and corresponding isomorphisms of Dynkin diagrams) of the different families of simple Lie algebras. Specifically, we have

Spin(1) = O(1)

Spin(2) = U(1) = SO(2) which acts on by double phase rotation

Spin(3) = Sp(1) = SU(2), corresponding to

Spin(4) = Sp(1) × Sp(1), corresponding to

Spin(5) = Sp(2), corresponding to

Spin(6) = SU(4), corresponding to

There are certain vestiges of these isomorphisms left over for n = 7, 8 (see Spin(8) for more details). For higher n , these isomorphisms disappear entirely.