The tangent space at the identity is a Lie algebra

By [F2] and [F3], $\mathfrak{g}=T_{e}G$ is an $n$-dimensional real vector space and hence is finite-dimensional.

Because the inverse of the linear isomorphism in [F6] is linear, $(au+bv{)}^L=a{u}^L+b{v}^L$. Bilinearity of the field bracket in [F7] and the definition [F4] therefore give $[au+bv,w{]}_G=a[u,w{]}_G+b[v,w{]}_G$, and similarly in the second variable.

Alternation of the field bracket in [F7] gives $[u,u{]}_G=[u^L,u^L{]}_e=0$. Thus the tangent bracket is alternating.

By [F4], the left-invariant extension of $[v,w{]}_G$ is $[v^L,w^L]$. Consequently the tangent Jacobi expression is the value at $e$ of $[u^L,[v^L,w^L]]+[v^L,[w^L,u^L]]+[w^L,[u^L,v^L]]$, which vanishes by the vector-field Jacobi identity in [F7].

Steps 1.1--1.4 verify every axiom in [F5], so $\mathfrak{g}$ is a finite-dimensional real Lie algebra. A Lie group is nonempty. If $n=0$, then $\mathfrak{g}=0$ and the bracket is the unique zero bracket; if $n=1$, alternation forces the bracket to vanish, consistently with the proof. No metric or nondegeneracy condition occurs, and the group is boundaryless by convention. The stated ${\mathrm{A}\mathrm{C}}_{\omega }$ is inherited through [F4], [F6], and the smooth-field meaning in [F7]; all algebraic transport is deterministic and adds no choice. The theorem verifies a structure rather than an iff.