Spin-independent axis-angle composition ($D^{(S)}(g_1{g}_2)=D^{(S)}(g_1)D^{(S)}(g_2)$)

The axis and angle of a product of rotations are properties of the group elements, independent of which spin representation acts on a state. Higher-spin generators do not obey the two-term Pauli exponential, but representations preserve the resulting rotation product. The central $2\pi$ rotation acts as $(-1{)}^{2S}I$.