A Lie Bracket Decomposition and Its Application to Flows on Symmetric Matrices

This paper is motivated by the problem of finding a canonical form for differential equations on symmetric matrices, that applies in particular to isospectral flows and spectrum-increasing flows. Such a canonical form is obtained using the general result; For any Lie algebra U and elements W,Z E U such that Z is ad-semisimple, there ixist unique A, B E U with W = [Z,A] + B such that A = {Z,C} for some C E U and [Z,B] + 0.