This paper deals with the striking fact that there is an essentially
canonical path from the $i$-th Lie algebra cohomology cocycle, $i=1,2,... l$,
of a simple compact Lie algebra $\g$ of rank $l$ to the definition of its
primitive Casimir operators $C^{(i)}$ of order $m_i$. Thus one obtains a
complete set of Racah-Casimir operators $C^{(i)}$ for each $\g$ and nothing
else. The paper then goes on to develop a general formula for the eigenvalue
$c^{(i)}$ of each $C^{(i)}$ valid for any representation of $\g$, and thereby
to relate $c^{(i)}$ to a suitably defined generalised Dynkin index. The form of
the formula for $c^{(i)}$ for $su(n)$ is known sufficiently explicitly to make
clear some interesting and important features. For the purposes of
illustration, detailed results are displayed for some classes of representation
of $su(n)$, including all the fundamental ones and the adjoint representation.