What is derivative? We’re familiar with the regular derivative as

Lie algebra arises from the study of Lie groups, which are groups that describe continuous symmetries (e.g., rotation, translation). It is the tangent space on each point in the Lie group. However, in terms of derivative, Instead of taking a derivative of a function, we look at how small transformations behave under commutation.

Note we are familiar is the canonical rotation, which is just rotation direction difference: