A semi-simple Lie algebra $L$ can by definition be decomposed into simple Lie algebras :
$L=L_1\oplus \ldots \oplus L_n $. Are these $L_i$ necessarily ideals of $L$?
2026-03-27 13:47:25.1774619245
Decomposition of semi-simple Lie algebra into simple lie algebra (or ideal?)
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Yes, the direct summands are ideals by definition, since a Lie algebra direct sum $L=L_1\oplus L_2$ is defined with the Lie bracket $[L_1,L_2]=0$. Hence we have $$[L_1,L]=[L_1,L_1\oplus L_2]=[L_1,L_1]\oplus[L_1,L_2]=L_1,$$ hence $L_1$ is an ideal. The same is true for $L_2$.