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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?

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    $\begingroup$ I don't know what definition is appropriate, but, whatever it is, one would have to give up on the sort of uniqueness one gets in the complex case. $\endgroup$
    – LSpice
    Sep 29, 2022 at 21:56

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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.

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