Lie Bracket

When a Lie Group is not commutative ($A\ast B\ne B\ast A$), then we end up having to account for this when we do Lie Algebra. This is because we want to do nice vector addition in the Lie Algebra, but if the Lie Group is not commutative, then we end up with error.

Core

$$X+Y=Y+X\text{(vectors commute!)}$$ $$g\ast h\ne h\ast g(\text{group elements often don’t commute})$$

Result

As a result, when you try to map from Lie Algebra to Lie Group with the exponential:

$$\exp (X+Y)\ne \exp (X)\cdot \exp (Y)(\text{for non-commutative Lie Groups})$$

So we need a formula that accounts for this distortion: $\exp (X)\cdot \exp (Y)=\exp \left(X+Y+\frac{1}{2}[X,Y]+\frac{1}{12}[X,[X,Y]]-\frac{1}{12}[Y,[X,Y]]+\cdots \right)$ Where $[X,Y]$ is the Lie Bracket and its a common function that deals with the distortion between moving in Lie Algebra and the Lie Group.