Relationship between $su(4)$, $so(4)$ and $su(2)\oplus su(2)$

Question

What is the relationship between the Lie algebras $su(4)$, $so(4)$ and $su(2)\oplus su(2)$ (if any)? I have read that $so(4)=su(2)\oplus su(2)$ but what is their relationship to $su(4)$?

Answer 1

Yes, $\mathfrak{su}(2)\bigoplus\mathfrak{su}(2)\simeq\mathfrak{so}(4,\Bbb R)$. This comes from the fact that, if you see $SU(2)$ as the group of quaternions with norm $1$ and if $\Bbb H$ is the space of quaternions, then the image of the map$$\begin{array}{ccc}SU(2)\times SU(2)&\longrightarrow&\operatorname{Aut}(\Bbb H)\\(q,r)&\mapsto&\left(\begin{array}{ccc}\Bbb H&\longrightarrow&\Bbb H\\h&\mapsto&qhr^{-1}\end{array}\right)\end{array}$$is isomorphic to $SO(4,\Bbb R)$ (it is the set of the norm-preserving automorphims of $\Bbb H$), and its kernel is discrete (it is equal to $\{\pm(e,e)\}$).

And $SO(4,\Bbb R)$ is a strict subgroup of $SU(4)$.