Triangular Derivations of k[X1, X2, X3, X4]

It is shown that if k is a field of characteristic zero, then the kernel of any triangular k-derivation of k[X1, X2, X3, X4] is finitely generated as a k-algebra. This is obtained as a corollary of a more general result concerning triangular R-derivations of R[X, Y, Z] for certain rings R.