  Let X be a normal variety endowed with an algebraic torus action. An additive
group action $\alpha$ on X is called vertical if a general orbit of $\alpha$ is
contained in the closure of an orbit of the torus action and the image of the
torus normalizes the image of $\alpha$ in Aut(X). Our first result in this
paper is a classification of vertical additive group actions on X under the
assumption that X is proper over an affine variety. Then we establish a
criterion as to when the infinitesimal generators of a finite collection of
additive group actions on X generate a finite-dimensional Lie algebra inside
the Lie algebra of derivations of X.
