  A "Composition map" is constructed, leaning heavily on earlier work [Y.
Fittouhi and A. Joseph, Parabolic adjoint action, Weierstrass sections and
components of the nilfibre in type $A$, Indag Math. and Y. Fittouhi and A.
Joseph, The canonical component of the nilfibre for parabolic adjoint action,
Weierstrass sections in type $A$, preprint, Weizmann, 2021]. It defines a
composition tableau which recovers the "canonical" Weierstrass section $e+V$
described in the first paper above. Moreover \textit{without reference to this
earlier work}, it is then shown that $e+V$ is indeed a Weierstrass section.
This results in a huge simplification. Moreover one may read off from the
composition tableau the "VS quadruplets'' of the second of the above papers,
thereby describing the "canonical component" of the nil-fibre in which $e$ lies
but does not of itself determine.
