Coroots, Killing Form and Langlands Dual Group

1. Given a semisimple Lie algebra T, we can create an adjoint representation.
2. Then, the Killing form is a bilinear form B that eats two elements of the Lie algebra and outputs a complex number. B(X,Y) = Trace(adj_X, adj_Y) where X and Y in T. adj_A is defined as adj_A = [A,Z] where A and Z is some element of T. adj_A therefore acts on vector Z from T and we can call it an operator. Composing two such operators and then taking the trace is the definition of the Killing form. So adj_A(Z) = [A,Z], adj_B(Z) = [B,Z] and adj_A(Z) composed with adj_B(Z) is [A,[B,Z]]. The composed action (adj_A composed with adj_B) on a test vector Z is [A,[B,Z]]. The trace of this composed action (which doesn’t involve Z) is the Killing form definition.
3. Remember, a root L of a Lie algebra T and Cartan Subalgebra (CSA) S means that [H,X] = L(H)*X, where X in T, H in CSA and L(H) is a number. You can say that L is a vector with components L(H_i) (not a vector of T).
4. If L(H) = B(H_L,H) for all H in the CSA, then we say that H_L is the unique CSA element that corresponds to the root L.
5. Now, if you plug H_L twice into the Killing form (B(H_L,H_L)), you would get a number. A coroot C_L of the root L is defined as 2*H_L/B(H_L,H_L).