This is really a simple naive question.
We know Levi-Civita symbol $\epsilon^{abc}$ has a nice property: https://en.wikipedia.org/wiki/Levi-Civita_symbol#Proofs $$ \epsilon^{abc} \epsilon^{ade}=\delta^{b,d}\delta^{c,e}-\delta^{b,e}\delta^{c,d} $$
$\delta^{i,j}$ is the delta function between $i,j$. Here $a$ is summed over as $\sum_{a}$.
Is there any Lie algebra structure constant formula similar to the above for $$f^{abc} f^{ade},$$ has any simplified form? Say, $$f^{abc} f^{ade}=\delta^{b,d}\delta^{c,e}-\delta^{b,e}\delta^{c,d} + ...?$$
If so, can you give a Reference or suggest where to find a proof (if you do not want to prove it).
We can consider for example semi-simple Lie algebra?
We can also consider compact semi-simple Lie group.
see another question too: Symmetric tensor of Lie algebra of $su(N)$