I have a question on principal 3-dimensional subalgebras. I know such concept in the complex case, but I'm not sure of the definition in the real case. Also, if the definition of the principal 3-dimensional subalgebra of a real form of a complex simple Lie algebra $\mathfrak g$ were that one whose complexification gives the principal 3-dimensional subalgebra of $\mathfrak g$, the existence and uniqueness are clear if the real form is the compact real form?
1 Answer
If your definition is a subalgebra $\mathfrak{s}$ which complexifies to a principal TDS you immediately have two distinct types $\mathfrak{s} \cong \mathfrak{su}_2$ or $\mathfrak{s} \cong \mathfrak{sl}(2,\mathbb{R})$. The Jacobson-Morozov Theorem should guarantee the existence of the latter in the noncompact case (we just need a regular nilpotent element in $\mathfrak{g}_\mathbb{R}$).
As to uniqueness for each type separately, I believe they must break into orbits. Focusing on the $\mathfrak{sl}(2,\mathbb{R})$ type, we note that the regular nilpotent elements certainly come in multiple orbits and as soon as there are more than two orbits of these (since there are two nilpotent orbits in $\mathfrak{sl}(2,\mathbb{R})$) that forces the TDS's to come in disjoint orbits as well.