Killing form

The Killing form or Cartan-Killing form(wikipedia), named after the mathematician Wilhelm Killing, is an invariant bilinear form $$(\_,\_)$$on a Lie algebra $$\mathfrak{g}$$ (with its defining vector space structure), defined on every pair $$(x,y)$$ of elements in $$\mathfrak{g}$$ as the trace $$(\_,\_) := tr(ad(x)ad(y))$$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:
 * $$([x,y],z) = (x,[y,z])$$ 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.

on paper

 * J.E.Humphreys, Introduction to Lie algebras and representation theory,ISBN 9780387900537, pp.21-
 * A.Knapp: Representation theory of semisimple groups, ISBN 0691090890, p.7