Can someone explain why the codimension of $\ker $ $\alpha $ is $1$ in $ H $, with complement $ Fh_\alpha $?
Is this because $ h_\alpha $ when $ \alpha $ is simple is part of the dual basis to the root space $ H^*$?
Where: $ H $ is the Cartan of a Lie algebra $ L $ (I don't know if it has to be semisimple)
$\alpha $ is a root not necessarily simple
$ h_\alpha=[{x_\alpha}{y_\alpha}]$ for $ x_\alpha $ $\in L_\alpha $ and $ y_\alpha$ $\in L_{-\alpha} $.
And $ F $ is the base field.