Lie Algebra = Generators + Commutation Relations. That’s it.

So imagine I have a Lie group G. By definition, a Lie group has infinite elements. Out of these infinite number of elements, there are 10 independent generators. Meaning, with these 10 elements, I can "get to" every other element of G. The set of these 10 elements, along with their commutation relations, [X,Y] = XY - YX, where X and Y are two of the generators, form the Lie algebra of this Lie group G. We say that the Lie algebra is 10 dimensional. For example, SO(3) is the group of all rotations in 3D space. It has 3 generators (infinitesimal rotations). So the Lie algebra of the Lie group SO(3) is a 3 dimensional algebra.

We also say that the tangent space at the identity element e is 10 dimensional. There are 10 different independent directions you can “go in”.