How often is a torus in a compact Lie group nullhomologous? Minor nomenclature question: 
What is the standard name for the ring structure induced on the homology of an H-space by its multiplication? I've seen "homology ring" and "Pontrjagin ring."
Hopefully deeper questions:
In this paper (on pp.  1117 & 1134), Samelson states (as best I can understand through Google Translate) that 
if a connected subgroup $U$ of a compact Lie group $G$ is not nullhomologous, then 


*

*the inclusion $U \hookrightarrow G$ induces an injection in rational homology, and

*the rational homology rings of $G$ and the product manifold $U \times (G/U)$ are isomorphic.
It occurs to me that one could use this to understand the (co)homology of homogeneous spaces $G/U$ (or alternately, assuming one knows something about homogeneous spaces, to understand the homology of $G$ in terms of homogeneous spaces and tori). How feasible is this strategy?
When are subgroups nullhomologous? I've never really thought of that before. I'm especially concerned about tori. My thinking is that the cohomology ring of a compact Lie group is an exterior algebra; by Poincaré duality, I would expect tori to generally be nullhomologous, then, because $\dim H_1(G)$ will tend to be too small to support large tori. Is that right? 
Are there illuminating examples? How should I think about this?
 A: The story is this: any compact, connected Lie group $G$ admits a finite, central cover of the form $\tilde G = A \times K$ for $A$ a torus and $K$ a simply-connected Lie group (hence a product of simply-connected simple Lie groups). This can be expressed in the form of a short exact sequence which is also a fiber bundle, 
$$0 \to F \to A \times K \to G \to 1.$$
Note that $H_1 = \pi_1$ on groups, so that $\pi_1 S \to \pi_1 G$ is assumed injective. Applying the long exact sequence of a bundle to the display, one finds $\pi_1 A \cong \pi_1 A \times \pi_1 K$ is a finite-index subgroup of $\pi_1 G$. It follows that the preimage in $\pi_1 S$ of the image of $\pi_1 A \to \pi_1 G$ is of finite index. Take a finite connected cover $\tilde S \to S$ such that the image of $\pi_1 \tilde S$ lies in the image of $\pi_1 A$; then one can lift $S \to G$ to a map $\tilde S \to \tilde G$ which must again be a homomorphism, and the map it induces in $\pi_1$ is injective.
The homotopy behavior of this map
$$\tilde S \to A \times K$$
is determined by the component homomorphism $\tilde S \to A$ since $K$ is simply-connected, and this map is recoverable from $\pi_1(A \to S)$ by the observation that $\pi_1$ and $- \otimes_{\mathbb Z} \mathbb R$ are inverse equivalences between the categories of finite-dimensional tori and finitely-generated free abelian groups.
In case the group $U$ is not a torus, the question is a bit more complex, effectively breaking down into separate questions for semisimple and toroidal components of (some cover of) $U$, and the interesting part is when an inclusion $K \hookrightarrow G$ of simple Lie groups induces a surjection in cohomology. I am not aware where to find a complete answer to this question, although I'm sure it's settled at least modulo torsion. A series of examples appears in the third volume of Greub, Halperin, and Vanstone's tome, around p. 474.
