Let $\mathfrak g$ be a finite-dimensional, semisimple Lie algebra over $\mathbb C$. I would like to know which semisimple elements of $\mathfrak g$ belong to some subalgebra $\mathfrak{sl}_2 \subseteq \mathfrak g$.
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2Since the union of Cartan subalgebras is the set of semisimple elements, your previous question might me relevant, too? – Dietrich Burde Aug 23 '19 at 18:53
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I do see a connection, and the fact that I asked both questions is by no means an accident. Nevertheless I don't see if the answer to the previous question solves this one. – Blazej Aug 23 '19 at 20:11
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Not all. If $x$ is semisimple and contained in a subalgebra isomorphic to $\mathfrak{sl}_2$, then since every semisimple element of $\mathfrak{sl}_2$ is regular, $x$ is proportional to a standard Cartan element $h$. As $\mathfrak{g}$ is a finite-dimensional $\mathfrak{sl}_2$-representation, the eigenvalues of $ad(h)$ are integers, and hence the eigenvalues of $ad(x)$ are integer multiples of some $c \in \mathbb C$. This is not true of all semisimple elements.
The converse seems to follow from the proof of the Jacobson-Morozov theorem: if the eigenvalues of semisimple $ad(x)$ are integer multiples of some $c \in \mathbb C$, then $x$ is contained in an $\mathfrak{sl}_2$-triple. These notes I found by googling "Jacobson-Morozov" contain the necessary steps of the proof in Section 3.
Joshua Mundinger
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