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A Lie algebra over a <field> is a <vector space> $\mathfrak g$ with a bilinear operation $[x,y]$, called the <Lie bracket>, that is alternating and satisfies the <Jacobi identity>.

= Lie algebras
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Lie algebra

The <Adjoint representation of a Lie algebra> is
$$
\operatorname{ad}:\mathfrak g\to\mathfrak{gl}(\mathfrak g),
\qquad \operatorname{ad}_X(Y)=[X,Y].
$$
It is linear. The <Jacobi identity> gives
$$
[\operatorname{ad}_X,\operatorname{ad}_Y]Z
=[X,[Y,Z]]-[Y,[X,Z]]=[[X,Y],Z]
=\operatorname{ad}_{[X,Y]}Z,
$$
so it is a <Lie algebra representation>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 302 / 3 / a / Solution

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The Jacobi identity is
$$
[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0.
$$


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 Incoming links: Jacobi identity