adjoint representation
A Lie algebra always carries one canonical representation built from nothing but its own bracket: let each element act on the whole algebra by bracketing with it. This self-action, the adjoint representation, is how the algebra sees itself, and it is the lens through which structural notions like nilpotence, the Killing form, and root spaces are all defined. The bracket is simultaneously the multiplication and the way the algebra acts on itself.
Define ad : L -> gl(L) by ad(x)(y) = [x, y]. The Jacobi identity is exactly the statement that ad is a Lie algebra homomorphism: ad([x, y]) = ad(x)ad(y) - ad(y)ad(x) = [ad(x), ad(y)]. So L acts on the vector space L itself, and ad(x) is a linear operator (a derivation, in fact) on that space for each x.
The kernel of ad is the center Z(L) — the elements bracketing trivially with everything — so for a Lie algebra with trivial center, ad embeds L faithfully into gl(L) as matrices. For a semisimple Lie algebra the center is zero and the adjoint representation is faithful; moreover every derivation of a semisimple Lie algebra is inner, i.e. of the form ad(x).
For sl(2) with basis (h, e, f), ad(h) acts diagonally with eigenvalues 0, 2, -2 on h, e, f respectively, since [h, h] = 0, [h, e] = 2e, [h, f] = -2f. So as a 3-by-3 matrix ad(h) = [0, 0, 0; 0, 2, 0; 0, 0, -2].
ad(h) on sl(2): a diagonal operator whose eigenvalues 0, 2, -2 are exactly the roots (and zero).
The adjoint representation is the Lie-algebra shadow of a Lie group acting on itself by conjugation; differentiating g x g^{-1} at the identity yields ad(X)(Y) = [X, Y].