Cartan subalgebra
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra h {\displaystyle {\mathfrak {h}}} of a Lie algebra g {\displaystyle {\mathfrak {g}}} that is self-normalising (if [ X , Y ] ∈ h {\displaystyle [X,Y]\in {\mathfrak {h}}} for all X ∈ h {\displaystyle X\in {\mathfrak {h}}} , then Y ∈ h {\displaystyle Y\in {\mathfrak {h}}} ). They were introduced by Élie Cartan in his doctoral thesis, and control the representation theory of a semi-simple Lie algebra g {\displaystyle {\mathfrak {g}}} over a field of characteristic 0 {\displaystyle 0} .