structure constants

Once you choose a basis for a Lie algebra, the bracket is determined entirely by what it does to pairs of basis vectors — everything else follows by bilinearity. The structure constants are just the bookkeeping numbers that record those basis brackets. They turn the abstract bracket into a concrete table of coefficients you can compute with by hand or feed to a computer.

Fix a basis e_1, ..., e_n of L. Because [e_i, e_j] is again an element of L, we can expand it in the basis: [e_i, e_j] = sum over k of c_{ij}^k e_k. The scalars c_{ij}^k in the field are the structure constants. Antisymmetry of the bracket forces c_{ij}^k = -c_{ji}^k, and the Jacobi identity imposes a quadratic constraint on them: for all i, j, l, the cyclic sum of c_{ij}^m c_{ml}^k over m vanishes.

Structure constants depend on the chosen basis — a change of basis transforms them by the corresponding tensor law — so they are not intrinsic invariants. But two Lie algebras are isomorphic exactly when some choice of bases makes their structure constants coincide, so the constants do carry the full isomorphism type.