How do I compute Levi-Civita connection of a left-invariant metric on a Lie group in a neighbourhood of $1$ by knowing only its Lie algebra and the metric form on it? I know it's possible because a Lie group is determined by its Lie algebra in some small neighbourhood of $1$, but I just found out that I forgot how to compute Levi-Civita connection in this case in practice :) Is there some nice formula for this?
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As per Alexei's suggestion, I'm making this an answer.
The Koszul formula is the usual technique for working out the Levi-Civita connection given a metric. In your particular case, if one restricts to left invariant vector fields, one can use the inner product on the Lie algebra and the Lie algebra structure to work out the Koszul formula.