Conformal Killing equation

In $d$ dimensions, the conformal Killing equation is

For a flat Conformal Killing vector field, let $\rho ={\partial }_{\mu }{\omega }^{\mu }$. Differentiate the Conformal Killing equation in three index orders, add two equations and subtract the third to obtain the displayed identity. Tracing gives $\Delta {\omega }_{\alpha }=(2-d){\partial }_{\alpha }\rho /d$, whose divergence implies $\Delta \rho =0$ for $d>1$. Taking the divergence of the original identity gives $(d-2){\partial }_{\beta }{\partial }_{\gamma }\rho =-{\delta }_{\beta \gamma }\Delta \rho$. Thus $\rho$ is affine and $\omega$ is at most quadratic when $d>2$.