  In this paper we continue our study of the geometric properties of full
symmetric Toda systems from \cite{CSS14,CSS17,CSS19}. Namely we describe here a
simple geometric construction of a commutative family of vector fields on
compact groups, that include the Toda vector field, i.e. the field, which
generates the full symmetric Toda system associated with the Cartan
decomposition of a semisimple Lie algebra. Our construction makes use of the
representations of the semisimple algebra and does not depend on the splitness
of the Cartan pair. It is very close to the family of invariants and
semiinvariants of the Toda system associted with $SL_n$, introduced in
\cite{CS}.
