From Humphrey's book, page 23:
In the second paragraph, it is stated that any ideal of $L_1$ is an ideal of $L$. How is this so?
Also, it is stated that this implies $L_1$ is semisimple. How is this so?
2026-04-06 11:38:26.1775475506
If $L$ is semisimple, how is this ideal of $L$ semisimple?
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The first question has already been answered before.
Concerning the other question, let $\mathfrak a$ be a solvable ideal of $L_1$. Then it is a solvable ideal of $L$. But $L$ is semisimple, which means that its only solvable ideal is $\{0\}$. So, $\mathfrak a=\{0\}$, which proves that $L_1$ is semisimple.