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- TL;DR Summary
- About solvable algebra

**Please, in the book of Introduction to Lie Algebras and Representation Theory J. E. Humphreys p.11, I have a question:**

Proposition.Let ##L## be a Lie algebra.

Proposition.

(a) If ##L## is solvable, then so are all subalgebras and homomorphic images of ##L##.

(b) If ##I## is a solvable ideal of ##L## such that ##L / I## is solvable, then ##L## itself is solvable.

(c) If ##I, J## are solvable ideals of ##L##, then so is ##I+J##.

Please, in proof, (b); how we get:

##\left(L^{(i)}\right)^{(j)}=L^{(i+j)} \text { implies that } L^{(n+m)}=0##

**Thanks in advance,**