  In this paper, using Bourbaki's convention, we consider a simple Lie algebra
$\mathfrak g\subset\mathfrak g\mathfrak l_m$ of type B, C or D and a parabolic
subalgebra $\mathfrak p$ of $\mathfrak g$ associated with a Levi factor
composed essentially, on each side of the second diagonal, by successive blocks
of size two, except possibly for the first and the last ones. Extending the
notion of a Weierstrass section introduced by Popov to the coadjoint action of
the truncated parabolic subalgebra associated with $\mathfrak p$, we construct
explicitly Weierstrass sections, which give the polynomiality (when it was not
yet known) for the algebra generated by semi-invariant polynomial functions on
the dual space $\mathfrak p^*$ of $\mathfrak p$ and which allow to linearize
the semi-invariant generators. Our Weierstrass sections require the
construction of an adapted pair, which is the analogue of a principal
$\mathfrak s\mathfrak l_2$-triple in the non reductive case.
