This might be a very trivial question for those of you well versed in Lie algebras, and if so I apologise… Any help is very much appreciated!
Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$. If I have an $Ad_G$-invariant metric on $G$, it is my understanding that this gives rise to an $ad_{g}$-invariant inner product on $\mathfrak{g}$, let's call this $\langle . , . \rangle$. Is this inner product also invariant under $Ad_G$, so that
$\langle [w,u] , v \rangle+ \langle u , [w,v] \rangle = 0 \quad$ for $u,v,w\in \mathfrak{g}$
and
$\langle [w,u] , [w,v] \rangle = \langle u , v \rangle \quad$?
Or have I missunderstood something? It almost seems too simple to be true…
Best Answer
The first identity is indeed coming from an ad-invariant metric, see the paper by G. Ovando Lie algebras with ad-invariant metric for a survey, with some low-dimensional examples. There are also several related questions here at this site:
Which Lie groups have Lie algebras admitting an Ad-invariant inner product?
There is a one-to-one correspondence between (a) left-invariant metrics on a connected simply connected Lie group $G$ and (b) Ad-invariant scalar products on the Lie algebra $\rm{Lie}(G)$.
Edit: The second identity, for all $w$, would imply $\langle u,v\rangle=0$, by taking $w=0$.