---- > [!definition] Definition. ([[Cartan subalgebra]]) > Let $\mathfrak{g}$ be a [[semisimple Lie algebra|semisimple]] [[Lie algebra]] over $\mathbb{C}$. A [[Lie subalgebra]] $\mathfrak{t}$ is said to be **toral** if $\mathfrak{t}$ is [[abelian Lie algebra|abelian]] and any $x \in \mathfrak{t}$ is [[semisimple element of a semisimple Lie algebra|semisimple as an element]] of $\mathfrak{g}$. > > A toral subalgebra not contained in a bigger one is called a **maximal toral subalgebra** or, more commonly, a **Cartan subalgebra (CSA)**. > Since the zero subalgebra is toral, it is clear that CSAs exist. In fact, a CSA is nonzero when $\mathfrak{g}$ is: if not, then every element of $\mathfrak{g}$ would be [[nilpotent element of a semisimple Lie algebra|nilpotent]], [[Engel's Theorem|whence]] $\mathfrak{g}$ is [[derived and central series of a Lie algebra|nilpotent]], [[The Cartan-Killing Criterion|contradicting]] [[semisimple Lie algebra|semisimplicity]]. example diagonal matrices from notes > [!note] Remark. > The name 'toral' comes from Lie groups — explain. ^note ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```