Example of a parallelizable smooth manifold which is not a Lie Group

Question

All the examples I know of manifolds which are parallelizable are Lie Groups. Can anyone point out an easy example of a parallelizable smooth manifold which is not a Lie Group? Are there conditions on a parallelizable smooth manifold which forces it to be a lie group?

Answer 1

Any closed, orientable $3$-manifold $X$ is parallelizable. On the other hand, Lie groups have $\pi_2$ trivial, and there are a lot of such $X$ that don't. (It's not trivial to find them, though. Note that any such $X$ must have $\pi_1 X\not = 0$ by Poincare duality, and aspherical (e.g., hyperbolic) $X$ have $\pi_2 X = 0$ as well. $3$-manifolds are weird.)

(The easiest way to prove the first assertion is to note that the only obstructions to the triviality of $TX$ are the Stiefel-Whitney classes; and $w_1 = 0$ by orientability and $w_2 = 0$ by the Wu formula, since we're conveniently in low dimension. The second is practically a folklore theorem, but it's not that hard to prove directly either via Morse theory or minimal models.)

Answer 2

The classical example is $S^7$. This comes from the octonions, which are non-associative, so the unit octonions don't form a Lie group.