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I fail to see how the Levi decomposition of the $\mathfrak g$ should be related to the Levi decomposition of $\mathfrak h$. By identifying $\mathfrak h$ with a Lie subalgebra in $\mathfrak g$ it is clear that the Levi decomposition of the whole algebra does not bring much information on the LEvi decomposition of the subalgebra (as the example of $\mathfrak g=\mathfrak{gl}_n$ shows...)