= Casimir splitting of a trivial quotient
{c}
= Splitting of a trivial quotient for a semisimple Lie algebra
{synonym}
In a finite-dimensional <short exact sequence> $0\to N\to E\to\mathbb C\to0$ of representations of a complex <semisimple Lie algebra>, the generalized zero <eigenspace> of the <Casimir operator> maps onto the trivial quotient. All its irreducible <composition factors> have zero <Casimir eigenvalue>, so they are trivial. The action is therefore strictly upper triangular and has solvable image; because a <semisimple Lie algebra> is a <perfect Lie algebra>, that image is zero. Every lift in this generalized zero <eigenspace> is invariant, giving a <split short exact sequence>.
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