Various definitions

Consider a Lie algebra g over a acreage K. Every aspect x of g defines the adjoint endomorphism ad(x) (also accounting as adx) of g with the advice of the Lie bracket, as

\textrm{ad}(x)(y) = [x, y].\

Now, admitting g is of bound dimension, the trace of the agreement of two such endomorphisms defines a symmetric bilinear form

B(x, y) = trace(ad(x)ad(y)),

with ethics in K, the Killing anatomy on g.