at.algebraic topology - Whitehead products on manifolds

I guess that by the Whitehead Lie algebra, you mean the homotopy group Lie algebra $pi_*(Omega X)simeq pi_{*-1}(X)$ maybe tensored by the reals $R$. In that case there is a theorem of Felix-Halperin-THomas, called the dichotomy theorem which tells you that either this Lie algebra is finite-dimensional (and the space is said to be "elliptic"), or it is very big in the sense that the ranks of $pi_k(X)$ grows exponentially with k (and the space is then called "hyperbolic"). If the Euler characteristic of the manifold is negative then the space is always hyperbolic/ Moreover when the space is hyperbolic the Whitehead Lie algebra is very far from being abelian: actually its radical is finite dimensional. Therefore any manifold with negative euler characteristic has an non abelian infinite dimensional homotopy Lie algebra.

To generalize what Ryan says, actually any connected sum of two simply connected manifolds $M$ and $N$ is hyperbolic unless the cohomology of both $M$ and $N$ are truncatated polynomial algebras on a single genrator (like the sphere or $CP(n)$). In particular the connected sum of 3 or more closed manifolds not having the rational homotopy type of a sphere is hyperbolic.

Another example of a non abelian Whitehead Lie algebra but finite dimensional, is the one associated to a manifold $M$ obtained as an $S^5$-bundle with base $S^3times S^3$ and where the euler class of the bundle is the fundamental class of the base (or any non zero multiple of it). In that case the Whitehead rational Lie algebra $pi_*(M)otimes Q$ is of dimension $3$ with basis $x,y,[x,y]$ where $x$ and $y$ are in degree $3$ and $[x,y]$ is in degree $5$. Thus this manifold M is elliptic. Interestingly enough, the cohomology algebra of M is isomorphic to that of the connected sum $W$ of two copies of $S^3times S^8$, but $W$ is hyperbolic.