Lie algebra of a Lie group literature

Question

I'm currently writing my thesis and want to use the concept of Lie-algebras. I explained everything with the definitions of a lie-algebra that I had of an old lecture. There the lie-algebra of a lie-group is defined as the tangent space at the identity element. I understand that there is an isomorphism between this tangent-space and the set of left-invariant vector-fields but I do not want to make things more complicated and just state it like this. The script does not have literature advice and in the standard differential geometry literature that I know the definition is different.

Can anyone advice me a citable document?

Answer 1

I like "Differential Geometry and Lie Groups A Computational Perspective" here.

See definition $18.6$

Answer 2

"Can anyone advice me a cite-able document?" I am unsure about what kind of answer/document you want.

Like this one? http://courses.theophys.kth.se/SI2320/luku4.pdf

at side $45$:

Since the left (right) translation is bijective, a left (right) invariant vector field isuniquely determined by giving its value at a single point, at the identity, say. Thus as a vector space, the space of left invariant vector fields can be identified as thetangent spaceTeGat the neutral elemente $\in G$.