The inner product in Cartan matrix

Question

Let $\Delta=\{e_i-e_{i+1} \}_{i=1}^{i=n}$ be a simple root system of $sl(n,\mathbb C)$, where $e_i\in H^*$ be such that $e_i(X)=$ the $i^{th}$ entry of $X$, and $H\leq sl(n,\mathbb C)$ consists of diagonal entries.

Now by definition Cartan matrix is given by $$A_{i,j}:= \frac{2\langle\alpha_i,\alpha_j\rangle}{\langle\alpha_i,\alpha_i\rangle}$$ But I don't understand what is the inner product $\langle,\rangle$?

Answer 1

The inner product is as simple as it gets: $$\langle e_i,e_j\rangle=\delta_{ij}$$ In general the inner product is given by the Killing form.