  We prove that every smooth affine variety of dimension $d$ embeds into every
simple algebraic group of dimension at least $2d+2$. We do this by establishing
the existence of embeddings of smooth affine varieties into the total space of
certain principal bundles. For the latter we employ and build upon parametric
transversality results for flexible affine varieties due to Kaliman. By
adapting a Chow-group-based argument due to Bloch, Murthy, and Szpiro, we show
that our result is optimal up to a possible improvement of the bound to $2d+1$.
In order to study the limits of our embedding method, we use rational homology
group calculations of homogeneous spaces and we establish a domination result
for rational homology of complex smooth varieties.
