= Linear Jahn-Teller selection rule
{title2=$\Gamma_Q\subset\operatorname{Sym}^2\Gamma_{\rm el}$}
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 $\operatorname{Sym}^2T_1$ and $\operatorname{Sym}^2T_2$ 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.
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