Let \({\mathfrak {p}}\) be a proper parabolic subalgebra of a simple Lie algebra \({\mathfrak {g}}\) . Writing \({\mathfrak {p}}=\mathfrak r\oplus \mathfrak m\) with \(\mathfrak r\) being the standard Levi factor of \({\mathfrak {p}}\) and \(\mathfrak {m}\) the nilpotent radical of \({\mathfrak {p}}\) , we consider the Inönü-Wigner contraction \(\widetilde{{\mathfrak {p}}}\) of \({\mathfrak {p}}\) with respect to this decomposition : this is the Lie algebra which is the semi-direct product \(\mathfrak r\ltimes \mathfrak {m}^a\) , where \(\mathfrak {m}^a\) is an abelian ideal of \(\widetilde{{\mathfrak {p}}}\) , isomorphic to \(\mathfrak {m}\) as an \(\mathfrak r\) -module. The study of the algebra of symmetric semi-invariants \(Sy\bigl (\widetilde{{\mathfrak {p}}}\bigr )\) in the symmetric algebra \(S\bigl (\widetilde{{\mathfrak {p}}}\bigr )\) of \(\widetilde{{\mathfrak {p}}}\) under the adjoint action of \(\widetilde{{\mathfrak {p}}}\) was initiated in Fauquant-Millet (2025), wherein a lower bound for the formal character of the algebra \(Sy\bigl (\widetilde{{\mathfrak {p}}}\bigr )\) was built, when the latter is well defined. Here in this paper we build an upper bound for this formal character, when \({\mathfrak {p}}\) is a maximal parabolic subalgebra in a simple Lie algebra \({\mathfrak {g}}\) in type B, whose Levi subalgebra is associated with the set of all simple roots without a simple root of even index, using Bourbaki notation (we call this case the even case). We show that both bounds coincide. This provides a Weierstrass section for \(Sy\bigl (\widetilde{{\mathfrak {p}}}\bigr )\) and the polynomiality of \(Sy\bigl (\widetilde{{\mathfrak {p}}}\bigr )\) follows. As a by-product, we obtain that the derived subalgebra \(\widetilde{{\mathfrak {p}}}'\) of \(\widetilde{{\mathfrak {p}}}\) is nonsingular that is, the set of regular elements in the dual space of \(\widetilde{{\mathfrak {p}}}'\) is big.
