I read here that a simple coroot $\alpha^{(i)}$ corresponds to a Cartan subalgebra element $H^i$ and don't understand why this should be the case.
Roots are the weights of the adjoint representation:
$$ [H_i , E_\alpha] = \alpha_i E_\alpha$$
where $H_i$ are the Cartan generators and $E_\alpha$ the other generators, i.e. the eigenvectors of the Cartan generators in the adjoint representation.
For another eigenvector we have, of course
$$ [H_i , E_{\alpha^{(2)}}] = \alpha_i^{(2)} E_{\alpha^{(2)}}$$
Therefore the superscript denotes the different roots and the subscript the corresponding components.
Each root $\alpha^{(k)}$ therefore corresponds to a non-Cartan generator. The simple roots are a linearly independent subset of the roots that can be used as a basis for the root space. For simplicity, let's assume we are dealing only with simply laced algebras, which means the roots are self-dual. Therefore the coroots coincide with the roots.
How does this fit together with the statement above that each simple coroot corresponds to a Cartan generator?