I am looking for an algorithm that can be used to determine Casimir invariants (of any order preferably) of any Lie algebra, starting from a specification of the commutation relationships of the algebra (which have been determined from a physical system). Several sources (e.g. [1,2]) map this problem onto the solution of a set of coupled partial differential equations, of the form
$$
C_{ij}^k a_k \frac{\partial f}{\partial a_j} = 0
$$
where the $C_{ij}^k$ are the structure constants of the Lie algebra, and homegeneous polynomial solutions $f$ of the $a_i$ (properly symmetrised) give the Casimir invariants.
How would one go about solving these nonlinear partial differential equations for an arbritary Lie algebra? For small algebras it is reasonable to write out each differential equation and solve by hand, but the algebras I am interested in have $>10$ elements and this quickly becomes tiresome and error-prone.
I suspect that diagonalising the $C_{ij}^k$ might help, but cannot quite see all the way through how to get to a generic solution method.
1 Alshammari, Isaac, Marquette, J. Phys. A, A differential operator realisation approach for constructing Casimir operators of non-semisimple Lie algebras
2 Chaichian, Demichev, Nelipa, Commun. Math. Phys., The Casimir Operators of Inhomogeneous Groups
Best Answer
Try the following reference: Simple evaluation of Casimir invariants in finite-dimensional Poisson systems by B. Hernández-Bermejo and V. Fairén, published in Physics Letters A, Volume 241, Issue 3, 27 April 1998, Pages 148-154.
In their notation you have $$J^{ij} = \sum_{k=1}^n c^{ij}_kx^k,$$ a linear (Lie-Poisson) structure.