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Take any Lie group irrep $R$ with $R\otimes{R}=R_1+R_2+R_3+R_4+1$. (Of course you can take the defining dimension of any member of the $E_8$ family (in that case $R_1=R$), but, say, $1\lambda_8$ of $D_8$ works as well. We take this as actual example: $128^2=6435+8008+1820+120+1$.
And now for my magic trick. Abracadabra! $3*(128+2)*128*6435*8008*1820*120=749548800^2$.
I see doubt in the auditorium, let's repeat with $E_8$: $3*(248+2)*248*248*3785*27000*30380=12108600000^2$.
For the members of the $E_8$ family you can easily check the truth of the statement directly (that the RHS is a square), but it holds as well for all the others, like $1\lambda_4(A_7)$ etc.
Since I used magic to derive it, that's cheating - do you have a math proof instead (via the Vogel plane or suchlike)? I can't believe this neat property went unnoticed yet...

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Notabene: For any point on the Vogel plane (i.e. R being the adjoint) the property is obvious indeed (just plug in the known dimensions and check that the formula gives a square). For the rest you still need magic :P – Hauke Reddmann May 27 '14 at 12:36

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