Trace of product of two linear transformations is independent of their order

Statement
Suppose $$V$$ is a finite-dimensional vector space over a field $$F$$. Suppose $$A,B \in \operatorname{End}(V)$$. Then, the trace of the composite $$AB$$ is the same as the trace of the composite $$BA$$.

Applications

 * Trace of product of linear transformations is invariant under cyclic permutations
 * Killing form is symmetric
 * Associativity-like relation between Killing form and Lie bracket

Similar facts

 * Characteristic polynomial of product of two linear transformations is independent of their order
 * Every power has trace zero iff nilpotent in characteristic zero
 * Nilpotent implies trace zero

Opposite facts

 * Minimal polynomial of product of two linear transformations may depend on their order