= Baker-Campbell-Hausdorff formula
{c}
{title2=BCH formula}
{wiki=Baker–Campbell–Hausdorff formula}
Solution $Z$ for given $X$ and $Y$ of:
$$
e^Z = e^X e^Y
$$
where $e$ is the <exponential map>.
If we consider just <real number>, $Z = X + Y$, but when X and Y are <non-commutative>, things are not so simple.
Furthermore, TODO confirm it is possible that a solution does not exist at all if $X$ and $Y$ aren't sufficiently small.
This formula is likely the basis for the <Lie group-Lie algebra correspondence>. With it, we express the actual <group operation> in terms of the Lie algebra operations.
Notably, remember that a <algebra over a field> is just a <vector space> with one extra product operation defined.
Vector spaces are simple because <all vector spaces of the same dimension on a given field are isomorphic>, so besides the dimension, once we define a <Lie bracket>, we also define the corresponding <Lie group>.
Since a group is basically defined by what the group operation does to two arbitrary elements, once we have that defined via the <Baker-Campbell-Hausdorff formula>, we are basically done defining the group in terms of the algebra.