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I have just come across the following experimental fact. Let ${\mathfrak g}$ be a simple complex Lie algebra.

Fact: ${\mathfrak g}$ is a constituent of $S^2{\mathfrak g}$ if and only if ${\mathfrak g}$ is of type $A_n$ with $n\geq 2$.

The corresponding symmetric product in type $A_n$ is easy to write: $$X\circ Y = XY+YX-\frac{2}{n+1} {\mathrm{Tr}}(XY)Id.$$ However,I understand the opposite conclusion only on a case by case basis.

Question: is there a scientific explanation of the fact? Is there a special name/theory for Lie algebras with such commutative product? Does the fact admit an interpretation in terms of the Vogel plane?

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  • $\begingroup$ The last question has an obvious answer since the trivial rep is a constituent of $S^2{\mathfrak g}$, so this condition determines 2 out of 3 Vogel parameters. I am really asking about deeper, more meaningful interpretation. $\endgroup$
    – Bugs Bunny
    May 16, 2018 at 7:54
  • $\begingroup$ This seems related.. mathoverflow.net/questions/269648/… $\endgroup$
    – spin
    May 16, 2018 at 7:58
  • $\begingroup$ This is equivalent to the existence of an invariant of degree 3 (since ${\mathfrak g}$ is self-dual, so if there is a copy of it in $S^2{\mathfrak g}$ then the associated map ${\mathfrak g}\times S^2{\mathfrak g}\rightarrow {\mathfrak g}\times {\mathfrak g}^* \rightarrow {\mathbb C}$ is an invariant of degree 3). $\endgroup$
    – Paul Levy
    May 18, 2018 at 7:03

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