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I'm interested in the low dimensional homotopy type of spaces like $SU(n)$. I know that the Whitehead products in cohomology vanish for all Lie groups. Does this mean that the Postnikov invariants (ie. $k$-invariants) are all trivial? What is known about the Postnikov invariants of the homotopy types of classical Lie groups?

Essentially I am after a discrete characterization of principal $G$-bundles in low dimensions, like how principal $U(1)$-bundles are classified by $H^2(X,\mathbb{Z})$, principal $G$-bundles should be classified by some discrete data involving the homotopy type of $BG$.

Thanks for your attention.

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    $\begingroup$ Whitehead products usually are operations on homotopy groups. If all Postnikov invariants for a space are trivial, then this space is a product of Eilenberg-Mac Lane spaces. This is not true for Lie groups (such as $S^3$). However, rationally there is a decomposition which relates to the fact that the Postnikov invariants for Lie groups (or H-spaces) have finite order. $\endgroup$
    – Matthias Wendt
    Commented Dec 20, 2016 at 13:05
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    $\begingroup$ For a classical example of how to classify bundles which also involves identification of Postnikov invariants in terms of cohomology operations see Dold-Whitney: Classification of oriented sphere bundles over a 4-complex. Ann. Math. 69 (1959), 667-677. $\endgroup$
    – Matthias Wendt
    Commented Dec 20, 2016 at 13:09

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