Definition 2

Tag 017D, permanent. Cite it as [Bourbaki, Tag 017D].

Lie Groups and Lie Algebras, chapter VIII, § 11. Classes of nilpotent elements and sl2 -triplets

An element $h$ of $\mathfrak{g}$ is said to be simple if there exist $x, y\in \mathfrak{g}$ such that $(x, h, y)$ is an $\mathfrak{s}\mathfrak{l}_2$ -triplet in $\mathfrak{g}$ .

We also say that $h$ is the simple element of the $\mathfrak{s}\mathfrak{l}_2$ -triplet $(x, h, y)$ .

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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