Kernels of derivations of polynomial rings and Casimir elements

We propose an algorithm for the evaluation of elements of the kernel of an arbitrary derivation of a polynomial ring. The algorithm is based on an analog of the well-known Casimir element of a finite-dimensional Lie algebra. By using this algorithm, we compute the kernels of Weitzenb?ck derivation d(x _i ) = x _i−1, d(x ₀) = 0, i = 0,…, n, for the cases where n ≤ 6.