Some of the accidental isomorphisms of low dimensional Lie algebras have very important applications in physics. The theory of angular momentum makes use of the fact that $SO(3)\sim SU(2)$. Similarly, the isomorphism $SO(4)\sim SU(2)\times SU(2)$ can be used to greatly simplify the representation theory of the Lorentz group in 4 dimensions. Another example is $SU(4)\sim SO(6)$, which I believe is useful for $\mathcal{N}=4$ SYM theory, where $SU(4)$ is the R-symmetry.

I was just wondering if there are any physical examples that make use of the last accidental isomorphism $SO(5)\sim Sp(2)$.

This post imported from StackExchange Physics at 2015-02-27 11:31 (UTC), posted by SE-user asperanz