The Killing form or Cartan-Killing form(wikipedia), named after the mathematician Wilhelm Killing, is an invariant bilinear form ${\displaystyle (\_,\_)}$on a Lie algebra ${\displaystyle {\mathfrak {g}}}$ (with its defining vector space structure), defined on every pair ${\displaystyle (x,y)}$ of elements in ${\displaystyle {\mathfrak {g}}}$ as the trace ${\displaystyle (\_,\_):=tr(ad(x)ad(y))}$of the matrix product for the adjoint representation of x and y. For a simple Lie algebra, the invariant bilinear form is unique up to scaling. A Lie algebra is semi-simple if and only if its Killing form is non-degenerate.

invariance

The Killing form has an invariance (or associative) property:

• ${\displaystyle ([x,y],z)=(x,[y,z])}$ where x,y,z are elements in the algebra and the brackets [] are the Lie brackets

exercise

• Write out the killing form for sl2, with its usual generators e,f and h.

references