Lie Algebra
Definition. $\mathfrak{g} = T_e G$, the tangent space to a Lie Group at the identity, equipped with the bracket $[X, Y]$ (commutator, for matrix groups) — bilinear, antisymmetric, Jacobi identity.
Working examples. $\mathfrak{u}(1) = i\mathbb{R}$ (bracket zero — abelian); $\mathfrak{su}(2)$ = traceless anti-Hermitian $2\times2$ matrices, spanned by $\{i\sigma_k/2\}$, with $[\tfrac{i\sigma_a}{2}, \tfrac{i\sigma_b}{2}] = -\epsilon_{abc}\tfrac{i\sigma_c}{2}$.
Intuition. Infinitesimal group elements. The bracket measures noncommutativity to lowest order: $e^{tX}e^{tY}e^{-tX}e^{-tY} = e^{t^2[X,Y] + O(t^3)}$.
Physics anchor. Angular momentum operators = $\mathfrak{so}(3) \cong \mathfrak{su}(2)$. Where gauge fields take values: a Connection 1-Form is $\mathfrak{g}$-valued. For EM, $\mathfrak{u}(1)$ is 1-dimensional ⇒ one field $A_\mu$; for QCD, $\dim\mathfrak{su}(3) = 8$ ⇒ eight gluon fields. And $[\omega,\omega] = 0$ for abelian $\mathfrak{g}$ is the reason Maxwell is linear while Yang–Mills is not.
Related: Lie Group · Exponential Map · Adjoint Representation · Connection 1-Form · Module 03 - Lie Groups and Lie Algebras
Sources: Sternberg Ch. V; Woit (Lie algebra chapters).