Killing form

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In mathematics,
the Killing form, named after Wilhelm Killing, is a symmetric bilinear form
that plays a basic role in the theories of Lie groups and Lie algebras.

Lie group (pronounced /liː/ "Lee") is a group that is also a differentiable manifold.
A manifold is a space that locally resembles Euclidean space,
whereas groups define the abstract, generic concept of multiplication and
the taking of inverses (division).

Combining these two ideas, one obtains a continuous group where
points can be multiplied together, and their inverse can be taken.
If, in addition, the multiplication and taking of inverses are defined to be smooth (differentiable), one obtains a Lie group.

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The Killing form is an inner product on a finite dimensional Lie algebra g defined by

B(X,Y)=Tr(ad(X)ad(Y)) (1)
in the adjoint representation, where ad(X) is the adjoint representation of X. (1)
is adjoint-invariant in the sense that

B(ad(X)Y,Z)=-B(Y,ad(X)Z). (2)
When g is a semisimple Lie algebra, the Killing form is nondegenerate.

https://mathworld.wolfram.com/KillingForm.html