Linear Jahn-Teller selection rule (${\Gamma }_Q\subset {\mathrm{Sym}}^2{\Gamma }_{\mathrm{el}}$)

For a real nonmagnetic electronic multiplet, a vibrational coordinate can couple linearly through a real symmetric electronic operator only if its representation occurs in the symmetric square. Remove the fully symmetric component when asking about symmetry-lowering distortion. In $T_d$, both ${\mathrm{Sym}}^2{T}_1$ and ${\mathrm{Sym}}^2{T}_2$ equal $A_1+E+T_2$, so $E$ and $T_2$ modes are symmetry-eligible for either triplet; coupling strengths are not determined by this decomposition.