  In this paper we develop further the relation between conformal four-point
blocks involving external spinning fields and Calogero-Sutherland quantum
mechanics with matrix-valued potentials. To this end, the analysis of
\cite{Schomerus:2016epl} is extended to arbitrary dimensions and to the case of
boundary two-point functions. In particular, we construct the potential for any
set of external tensor fields. Some of the resulting Schr\"{o}dinger equations
are mapped explicitly to the known Casimir equations for 4-dimensional seed
conformal blocks. Our approach furnishes solutions of Casimir equations for
external fields of arbitrary spin and dimension in terms of functions on the
conformal group. This allows us to reinterpret standard operations on conformal
blocks in terms of group-theoretic objects. In particular, we shall discuss the
relation between the construction of spinning blocks in any dimension through
differential operators acting on seed blocks and the action of left/right
invariant vector fields on the conformal group.
