  A self-adjoint element in a finite AW*-factor is spectrally symmetric, if its
spectral measure under the quasitrace is invariant under the change of
variables $t\longmapsto -t$. We show that if $\mathcal{A}$ is an AW*-factor of
type II_1, a self-djoint element in $\mathcal{A}$, without full support, has
quasitrace zero, if and only if it can be written as a sum of at most three
commuting spectrally symmetric elements.
