Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. We can use the Killing form to identify $\mathfrak{h}$ and $\mathfrak{h}^*$ ($\phi\in \mathfrak{h}^*$ corresponds to $t_{\phi}\in \mathfrak{h}$), where $t_{\phi}$ satisfies $\phi(h)=\kappa(t_{\phi}, h)$ for all $h\in \mathfrak{h}$ ($\kappa$ is the Killing form on $\mathfrak{g}$). Since $\kappa$ is non-degenerate, it is easy to show that the above correspondence is injective. But how to show that the above correspondence is surjective? Why each $h\in \mathfrak{h}$ is of the form $t_{\phi}$ for some $\phi \in \mathfrak{h}^*$?
I think there are other ways to identify $\mathfrak{h}$ and $\mathfrak{h}^*$. I am reading a paper path description of type B q-characters. On page 3, line&nbps;2 of section 2, it is said that $\mathfrak{h}$ and $\mathfrak{h}^*$ can be identified by using the invariant inner product $\langle\, , \rangle$ on $\mathfrak{g}$ normalized in such a way that the square length of the maximal root equals $2$. What is the relation of this form and the correspondence above?
Let $\langle\, , \rangle$ be the form defined as above. How to compute $\langle\alpha_i, \alpha_i\rangle$ explicitly for all types ($\alpha_i$ are simple roots)? Thank you very much.