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[Disclosure: As of 1/17/23, a version of this has been posted on mathoverflow]

This may be a very naive question, but I'm having some trouble counting degrees of freedom for Lie algebraic structure constants. Given an $n$-dimensional Lie algebra, there are $\frac{n^2(n-1)}{2}$ degrees of freedom for the structure constants if one ignores the Jacobi identity. Accounting only for very obvious symmetries, the Jacobi identity seems to impose $\frac{n^2(n-2)(n-1)}{6}$ constraints. For $n>5$, there are far more of these than structure constants. How does one reconcile this? Of course, they are quadratic equations, not linear ones --- so it's not as simple as counting components. But intuitively, it feels like the system is heavily over-constrained. On the other hand, we know that solutions exist for $n>5$ (ex. $so(n)$). It may be that I'm massively overcounting the Jacobi identity equations or am missing something obvious. I'd appreciate any guidance on this. In particular, if someone could point me to a more meaningful notion of "degrees of freedom" for the structure constants, I'd really appreciate it.

Thanks in advance for your help!

Cheers, Ken

Torsten Schoeneberg
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Kensmosis
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    The situation is even worse in some sense, because structure constants related by a mere basis change correspond to isomorphic Lie Algebras --- so there even are fewer effective degrees of freedom. – Kensmosis Aug 22 '18 at 21:03
  • Revisiting this: My comment about basis changes is wrong since the structure constant eqs are basis independent. Also, I'm moving this question to mathoverflow. – Kensmosis Jan 17 '23 at 16:38

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