Highest weight modules lie below the top weight

Given: The Axiom of Choice, such $\mathfrak{g},\mathfrak{h}$, a chosen positive system, and a module $V$ generated by a highest weight vector $v$ of weight $\lambda$.

The Axiom of Choice is assumed; it enters through the root-space theory supplying [L1], [L2] and [L4] (The Axiom of Choice).

The chosen positive system gives the direct sum $\mathfrak{g}={\mathfrak{n}}^{-}\oplus \mathfrak{h}\oplus {\mathfrak{n}}^{+}$ with ${\mathfrak{n}}^{\pm }$ the span of the root spaces ${\mathfrak{g}}_{\pm \alpha }$, $\alpha \in {\mathrm{\Phi }}^{+}$ (Triangular decomposition, Positive and negative nilpotent subalgebras and the Borel).

For an ordered basis $x_1,\dots ,x_N$ of a finite-dimensional complex Lie algebra, the monomials $x_1^{a_1}\cdots x_N^{a_N}$ form a basis of its universal enveloping algebra (Poincaré–Birkhoff–Witt theorem); applied to $\mathfrak{g}$ with an ordered basis beginning with a basis of ${\mathfrak{n}}^{-}$, continuing with a basis of $\mathfrak{h}$ and ending with a basis of ${\mathfrak{n}}^{+}$, this gives $U(\mathfrak{g})=U({\mathfrak{n}}^{-})U(\mathfrak{h})U({\mathfrak{n}}^{+})$ as a linear span, and applied to ${\mathfrak{n}}^{-}$ it gives that the monomials in a basis of ${\mathfrak{n}}^{-}$ span $U({\mathfrak{n}}^{-})$.

The action of $\mathfrak{g}$ on $V$ extends to a unital action of $U(\mathfrak{g})$, and the subrepresentation generated by $v$ is $U(\mathfrak{g})v$ (Lie representations are U(g)-modules, Highest-weight vectors and modules).

If $x\in {\mathfrak{g}}_{\alpha }$ and $w\in V_{\mu }$, then $x\cdot w\in V_{\mu +\alpha }$ (Root vectors shift weights).

Every positive root is a nonzero nonnegative integral combination of the simple roots, each root space is one-dimensional, and the simple roots are linearly independent (Simple roots form a signed integral basis, Root spaces of a complex semisimple Lie algebra are one-dimensional).