  We found in 2016 a few results on the mathematical structure of the conformal
Killing differential sequence in arbitrary dimension $n$, in particular the
rank and order changes of the successive differential operators for $n=3,n=4$
or $n\geq 5$. They were so striking that we did not dare to publish them before
our former PhD student A. Quadrat (INRIA) could confirm them while using new
computer algebra packages that he developped for studying extension modules in
differential homological algebra. In the meantime, as a complementary result,
we found in 2017 the "missing link" justifying the doubts we had since a long
time on the origin and existence of Gravitational Waves in General Relativity.
In both cases, the main tool is the explicit computation of certain extension
modules for the classical or conformal Killing differential sequences. These
results therefore lead to revisit the work of C. Lanczos and successors on the
existence of a parametrization of the Riemann or Weyl operators and their
respective formal adjoint operators. We also provide an example showing how
these extension modules are depending on the structure constants appearing in
the Vessiot structure equations (1903), still not acknowledged after one
century even though they generalize the constant Riemannian curvature
integrability condition of L.P. Eisenhart (1926) for the Killing equations. The
present paper is written from a lecture gven at the recent 24 th conference on
Applications of Computer Algebra (ACA 2018) held in Santiago de Compostela,
Spain, june 18-22, 2018.
