 Solution

Put $\kappa =8\pi G/c^4$ and $T=T_{\nu }{}^{\nu }$. The contracted Bianchi identity and metric compatibility giveTaking the trace of the field equation in four dimensions gives $(1-4\alpha )R=-\kappa T$. For $\alpha \ne 1/4$, eliminate $R$:Ordinary covariantly coupled matter obeys stress-energy conservation, ${\nabla }_{\nu }{T}_{\mu }{}^{\nu }=0$. Its trace need not be constant, so the trace coefficient fixed by stress-energy conservation isThis restores the Einstein tensor and is the only coefficient compatible with arbitrary conventional matter without adding a new constraint or changing the matter conservation law. Other coefficients can work for specially restricted sources with constant trace, but do not give the same universal matter coupling.

The excluded case $\alpha =1/4$ must be treated before dividing: its trace equation requires $T=0$. If stress-energy is also conserved, its divergence equation requires ${\partial }_{\mu }R=0$. It can describe a restricted traceless-source sector with constant scalar curvature, not general matter such as ordinary dust. In particular it is not the general trace-free Einstein equation with source $T_{\mu \nu }-g_{\mu \nu }T/4$, which is a different equation.