= Lie algebra
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Like everything else in <Lie groups>, first start with the <matrix> as discussed at <Lie algebra of a matrix Lie group>{full}.
Intuitively, a <Lie algebra> is a simpler object than a <Lie group>. Without any extra structure, groups can be very complicated non-linear objects. But a <Lie algebra> is just an <algebra over a field>, and one with a restricted <bilinear map> called the <Lie bracket>, that has to also be <alternating multilinear map>[alternating] and satisfy the <Jacobi identity>.
Another important way to think about Lie algebras, is as <infinitesimal generators>.
Because of the <Lie group-Lie algebra correspondence>, we know that there is almost a <bijection> between each <Lie group> and the corresponding <Lie algebra>. So it makes sense to try and study the algebra instead of the group itself whenever possible, to try and get insight and proofs in that simpler framework. This is the key reason why people study Lie algebras. One is philosophically reminded of how <normal subgroups> are a simpler representation of <group homomorphisms>.
To make things even simpler, because <all vector spaces of the same dimension on a given field are isomorphic>, the only things we need to specify a <Lie group> through a <Lie algebra> are:
* the dimension
* the <Lie bracket>
Note that the <Lie bracket> can look different under different basis of the <Lie algebra> however. This is shown for example at <Physics from Symmetry by Jakob Schwichtenberg (2015)> page 71 for the <Lorentz group>.
As mentioned at <Lie Groups, Physics, and Geometry by Robert Gilmore (2008)> Chapter 4 "Lie Algebras", taking the <Lie algebra> around the identity is mostly a convention, we could treat any other point, and things are more or less equivalent.
Bibliography:
* https://physicstravelguide.com/advanced_tools/group_theory/lie_algebras#tab__concrete on <Physics Travel Guide>
* http://jakobschwichtenberg.com/lie-algebra-able-describe-group/ by <Jakob Schwichtenberg>
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