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I am currently reading J-P Serre's book: Complex Semisimple Lie Algebras. On page 12 we have the following:

Theorem 1: Let $g$ be a Lie algebra. If $x$ is regular, the nilspace of $ad_x$ is a Cartan subalgebra of $g$, and its dimension is equal to the rank of $g$.

Proof: First, we show that the nilspace of $ad_x$ is nilpotent (we already have from previous results that it is a subalgebra). By Engel's theorem, it is sufficient to prove that for every $y$ in the nilspace of $ad_x$, the restriction of $ad_y$ to the nilspace is nilpotent. Let $ad^1_y$ denote this restriction, and $ad^2_y$ the endomorphism induced by $ad_y$ on the quotient-space given by $g$ quotiented by the nilspace of $ad_x$...

And it continues. I don't understand the meaning of $ad^2_y$; I don't think I've encountered the terminology "the endomorphism induced by..." before. What is $ad^2_y$?

Matt
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1 Answers1

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For a Lie algebra $L=(V,[,])$, the adjoint operator ${\rm ad}(y)$ is defined by ${\rm ad}(y)(x)=[y,x]$. Then usually $ad_y^2={\rm ad}(y)^2$ is the square. However, here Serre defines it differently. Induced means, that its definition on the quotient space $L/I$ follows from ${\rm ad}(y)$. This works always, for groups, algebras, rings and so on:

Induced homomorphism by passing to the quotient

Dietrich Burde
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