Ideals of Lie algebras

DEFINITION 5.1   For a linear subspace $S,T$ of a Lie algebra $L$ , let us define denote by $[S,T]$ the following linear subspace of $L$ .

1. $$[S,T]=($$linear span of $$\{[x,y]; x\in S, y\in T\})$$

DEFINITION 5.2   Let $L$ be a Lie algebra over a field $k$ . A $k$ -linear subspace $\mathfrak{a}$ of $L$ is said to be an ideal of $L$ if

$$[x,y]\in \mathfrak{a}\qquad(\forall x\in L,\forall y\in \mathfrak{a})$$

holds. This clearly is equivalent to saying that

$$[L,\mathfrak{a}]\subset L$$

holds.

PROPOSITION 5.3   Let $\mathfrak{a}$ be an ideal of a Lie algebra $L$ . Then

1. $\mathfrak{a}$ is a sub $L$ -module (sub representation) of $L$ .
2. $L/\mathfrak{a}$ caries a natural structure of a Lie algebra.

PROOF.. As usual. $\qedsymbol$