rt.representation theory - Restriction map for Lie algebra/Lie group cohomology associated to a complex semisimple Lie algebra and a semisimple Lie-subalgebra

Let $mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra (or the corresponding Lie group). For definiteness, I'll take $mathfrak{g}$ to be of type $A_n$, that is, $mathfrak{g} = mathfrak{sl}_{n+1}(mathbb{C})$, but my question applies to semisimple Lie algebras of arbitrary Lie type. Consider the Dynkin diagram for $mathfrak{g}$. We can remove a node from the diagram to obtain a sub-diagram of type $A_{n-1}$. The sub-diagram corresponds to a copy of the Lie algebra $mathfrak{g}':=mathfrak{sl}_n(mathbb{C})$ sitting inside $mathfrak{g}$.

The structure of the Lie algebra cohomology rings $H^bullet(mathfrak{g},mathbb{C})$ and $H^bullet(mathfrak{g}',mathbb{C})$ are known, and are the same as the cohomology rings $H^bullet(G,mathbb{C})$ and $H^bullet(G',mathbb{C})$ for the corresponding complex Lie group. The computation of the cohomology rings is classical; for Lie algebras the computation is a result of Koszul.

In the specific case $mathfrak{g} = mathfrak{sl}_{n+1}(mathbb{C})$, we have $H^bullet(mathfrak{g},mathbb{C}) = Lambda(x_3,x_5,ldots,x_{2n+1})$, an exterior algebra on homogeneous generators of degrees $3,5,ldots,2n+1$. Then $H^bullet(mathfrak{g}',mathbb{C}) = Lambda(x_3,x_5,ldots,x_{2n-1})$. (For other Lie types, the cohomology ring is still an exterior algebra on homogeneous generators of certain odd degrees depending on the root system.)

The inclusion of Lie algebras $mathfrak{g}' rightarrow mathfrak{g}$ gives rise to a corresponding restriction map in cohomology: $H^bullet(mathfrak{g},mathbb{C}) rightarrow H^bullet(mathfrak{g}',mathbb{C})$.

Is the restriction map in cohomology map the "obvious'' map from $Lambda(x_3,x_5,ldots,x_{2n+1})$ to $Lambda(x_3,x_5,ldots,x_{2n-1})$, that is, the map that takes $x_i$ to $x_i$ for $1 leq i leq 2n-1$ and that takes $x_{2n+1}$ to zero? If so, can you provide a reference for this fact? For other Lie types, is the restriction map also the obvious map?

Edit: In response to the comments below, I think I should have phrased the question as follows:

Hopefully clarified version of question: Is there a choice of generators for the cohomology rings $H^bullet(mathfrak{g},mathbb{C})$ and $H^bullet(mathfrak{g}',mathbb{C})$ such that the restriction map in cohomology has the above described form?

I acknowledge that things could get messier for restriction maps in type $D_n$ and for types $E_6$, $E_7$, and $E_8$, because in those cases the degrees of the generators for the cohomolgoy ring aren't as well-behaved. But maybe something can still be said in general about the restriction map (e.g., when is it surjective?).