Proper twin-triangular Ga-actions on A⁴ are translations

An additive group action on an affine 3 -space over a complex Dedekind domain A is said to be twin-triangular if it is generated by a locally nilpotent derivation of A[y,z,t] of the form rd/dy+p(y)d/dz + q(y)d/dt, where r belongs to A and p,q belong to A[y] . We show that these actions are translations if and only if they are proper. Our approach avoids the computation of rings of invariants and focuses more on the nature of geometric quotients for such actions.