A note on "A generalization of Roberts' counterexample to the fourteenth problem of Hilbert by S. Kuroda"

In [4], Kuroda generalized Roberts’ counterexample \cite{Roberts} to the fourteenth problem of Hilbert. The counterexample is given as the kernel of a locally nilpotent derivation on a polynomial ring. We replace his construction of the invariant elements by a more straightforward construction and give a more precise form of invariant elements.