TITLE: Homotopy properties of Lie groups
QUESTION [10 upvotes]: Let $G$ be a real connected Lie group. I am interested in its special homotopy properties, which distinguish it from other smooth manifolds
For example

$G$ is homotopy equivalent to a smooth compact orientable manifold. In particular, Poincaré duality holds for $G$.
$\pi_1(G)$ is abelian, $\pi_2(G) = 0$

The impossibly perfect answer to my question is a list of properties that make up a complete homotopy characterization of Lie groups (that is, in every homotopy type (of smooth manifolds) with such properties, there exists a smooth manifold admitting the structure of Lie groups).
P.S. In this question, I am not interested in the homotopy properties of manifolds that distinguish them from other CW-complexes, for this see resp. question on MO

REPLY [8 votes]: A very useful fact is that every connected Lie group is rationally homotopy equivalent to the product of several odd dimensional spheres where the number of spherical factors is equal to the rank of the group. For example, $SU(n)$ is rationally equivalent to $S^3\times S^5 \ldots \times S^{2n-1}$ and $Sp(n)$ is rationally equivalent to $S^3\times S^7\times \ldots \times S^{4n-1}$. For simple Lie groups the first factor is always $S^3$ and there is only one $S^3$ in each simple group. Also, every simple $G$ has $\pi_3\cong\mathbb Z$.  For reference see for example the book "Topology of transitive transformation groups" by Onishchik.