Big Ad and little ad: how a group acts on its own algebra
By now you have the Lie algebra g of a group G as its tangent space at the identity, T_e G, with the bracket [X, Y] inherited from left-invariant vector fields, and you know the exponential map exp: g -> G sends a tangent direction to a one-parameter subgroup. The natural next question is: how does the group act on its own algebra? G acts on itself by conjugation, C_g(h) = g h g^(-1), and this fixes the identity e for every g. So its differential at e is a linear map g -> g, written Ad(g) = d(C_g)_e. This is the adjoint representation of the group.
For a matrix group this is wonderfully concrete: conjugation is genuinely matrix conjugation, and Ad(g)X = g X g^(-1), where X is a matrix in g. The map g -> Ad(g) is a homomorphism into GL(g), the invertible linear maps on the vector space g — that is exactly what 'representation' means: the group is faithfully (or at least homomorphically) shadowed by linear transformations. So the adjoint representation Ad: G -> GL(g) repackages the group's conjugation structure as honest linear algebra on a fixed vector space.
Now differentiate one more time. Ad is itself a smooth map from the group G into the linear group GL(g); its differential at the identity is a linear map from g into the Lie algebra of GL(g), which is just End(g), the algebra of all linear endomorphisms. Call this ad = d(Ad)_e: g -> End(g). The punchline — the single most important formula in the subject — is that ad(X) is precisely bracketing with X:
Ad(g) X = g X g^(-1) (group level, matrix case) ad(X) Y = [X, Y] (algebra level) Ad(exp(tX)) = exp(t * ad(X)) (the two are linked by exp)
Why ad(X) = [X, -] is not a coincidence
It is worth pausing on why bracketing falls out of differentiating conjugation, because it ties together everything from the earlier guides. Fix X and Y in g and look at Ad(exp(sX))Y as a curve in g. Differentiating at s = 0 gives ad(X)Y by definition. For a matrix group, exp(sX) Y exp(-sX) has s-derivative at 0 equal to XY - YX — the commutator. So the abstract bracket on g, which we introduced as the bracket of left-invariant vector fields, agrees with the plain matrix commutator XY - YX whenever G is a matrix group. The two notions of bracket are one and the same.
This explains the Jacobi identity in one line. Differentiating the relation Ad(g)[Y, Z] = [Ad(g)Y, Ad(g)Z] (which says conjugation respects the bracket) gives ad(X)[Y, Z] = [ad(X)Y, Z] + [Y, ad(X)Z]. In words: ad(X) is a derivation of the bracket — it obeys a Leibniz rule. Unwinding the definition of ad turns this exactly into the Jacobi identity [X,[Y,Z]] = [[X,Y],Z] + [Y,[X,Z]]. So Jacobi is not a strange axiom pulled from a hat; it is the infinitesimal shadow of the obvious fact that conjugation is an automorphism.
The Killing form: a canonical inner product made from the bracket
With ad in hand we can manufacture a bilinear form on g out of nothing but the bracket. Define B(X, Y) = trace(ad(X) ad(Y)). Since ad(X) and ad(Y) are linear maps g -> g, composing them and taking the trace gives a number, and B is visibly symmetric and bilinear. This is the Killing form, and it is the single most useful gadget for diagnosing the structure of a Lie algebra. Its great virtue: it is invariant, meaning B(Ad(g)X, Ad(g)Y) = B(X, Y), and infinitesimally B([Z,X], Y) + B(X, [Z,Y]) = 0 — the bracket is skew with respect to B.
Why care about a trace of nested brackets? Because Cartan's criterion turns the signature of B into a structural verdict. The Killing form is non-degenerate exactly when g is semisimple — has no nonzero abelian ideal. And B is negative-definite exactly when g is the Lie algebra of a compact semisimple group. So a single symmetric matrix, computable by hand for small algebras, tells you whether the group is compact and whether it decomposes into simple pieces. For su(n) the Killing form is negative-definite (compact); for sl(n, R) it is indefinite (non-compact but semisimple); for an abelian algebra it is identically zero (as far from semisimple as possible).
On a semisimple algebra the non-degenerate Killing form gives a canonical way to raise and lower indices, and — crucially — a bi-invariant Riemannian metric on a compact group, where B (up to sign) furnishes an Ad-invariant inner product on g that is then spread over G by left translation. That metric's geodesics through the identity are exactly the one-parameter subgroups, and its curvature is computable purely algebraically from the bracket. This is the bridge by which root systems and the whole classification of semisimple Lie algebras feed back into honest Riemannian geometry.
The Maurer-Cartan form: the group's geometry in one 1-form
Now flip the viewpoint from the algebra back to the whole group, but globally. At every point g in G the left translation L_g: G -> G is a diffeomorphism, and its differential lets you carry any tangent vector v in T_g G back to the identity. The Maurer-Cartan form omega is the g-valued 1-form on G defined by omega_g(v) = (dL_{g^(-1)})_g(v) — it reads off 'which left-invariant field does v point along, measured at the origin.' For a matrix group this is heartwarmingly simple: omega = g^(-1) dg, where dg is the matrix of coordinate differentials. So omega is a 1-form whose values are matrices in the Lie algebra g, and at each point it is a linear isomorphism T_g G -> g.
The whole content is one equation, the Maurer-Cartan equation: d omega + (1/2)[omega ^ omega] = 0, where [omega ^ omega] brackets the Lie-algebra values and wedges the form parts. For a matrix group you can verify it in two lines: differentiate omega = g^(-1) dg using d(g^(-1)) = -g^(-1) (dg) g^(-1), and the cross terms reorganize into exactly the bracket term. This identity is the integrability condition: it says the form omega is 'flat' in a precise sense, and it is the structural fingerprint that distinguishes the geometry of one group from another.
Putting the three together
These are not three separate topics; they are three faces of one structure, and the way to feel that is to compute with a tiny example. Take G = SU(2), the unit quaternions, whose Lie algebra su(2) is the 3-dimensional space of trace-free skew-Hermitian 2x2 matrices, spanned by i times the Pauli matrices, call them X_1, X_2, X_3, with [X_1, X_2] = 2 X_3 and cyclic.
- Adjoint: Ad(g) acts on su(2) as rotation; the homomorphism Ad: SU(2) -> SO(3) is exactly the famous two-to-one cover, since g and -g conjugate identically. So 'the adjoint representation of SU(2)' literally is the spin double cover you met in the matrix-group guide.
- Killing form: B(X, Y) = 4 trace(XY) here, which is negative-definite — the algebraic certificate that SU(2) is compact and simple. Scaled, it is the round metric on the 3-sphere that SU(2) already is.
- Maurer-Cartan: omega = g^(-1) dg expands in the basis X_1, X_2, X_3 into three real 1-forms sigma_1, sigma_2, sigma_3 satisfying d sigma_1 = -2 sigma_2 ^ sigma_3 (and cyclic) — these are the left-invariant coframe on S^3, the very forms used to write the Hopf fibration and the round metric ds^2 = sigma_1^2 + sigma_2^2 + sigma_3^2.
Read those three lines side by side and the circle closes: the conjugation action (Ad) differentiates to the bracket (ad), the bracket builds the invariant form (Killing), and the same bracket is the structure constant tying together the global coframe (Maurer-Cartan). One small group, SU(2), simultaneously is the 3-sphere, the spin cover of rotations, and the cleanest worked example of every formula in this guide. In the next guide we let the group act on other spaces — homogeneous spaces — and integrate over it with invariant measure, turning this algebra into representation theory and harmonic analysis.