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It is a classical result that the rational cohomology of $SO(n)$ is given by:

$$H^*(SO(2m); \mathbb{Q}) = \begin{cases} S(\beta_1, \dots, \beta_{m-1}, \alpha_{2m-1}) & n = 2m \\ S(\beta_1, \dots, \beta_m) & n = 2m+1 \end{cases}$$ where the $\beta_i$ have degree $4i-1$, $\alpha_{2m-1}$ has degree $2m-1$, and $S(V)$ is the free graded commutative algebra on the generators given by the graded vector space $V$. Since all the generators have odd degree, this happens to be an exterior algebra here.

Since $SO(n)$ is a (Lie) group, there is an induced Hopf algebra structure on $H^*(SO(n);\mathbb{Q})$, and the generators $\beta_i$ (and $\alpha_{2m-1}$ if applicable) happen to be primitive. But then, a free (graded) symmetric algebra on primitive generators is (graded) cocommutative, so in this case $H^*(SO(n))$ is cocommutative.

Now, like all Lie groups, $SO(n)$ is formal, and all its rational homotopy theory is contained in its cohomology ring. This is where things start to get muddy for me: if I understand correctly, the coproduct on $H^*(SO(n))$ should be a model for the product on $SO(n)$. Since $H^*(SO(n))$ is cocommutative, then the product on $SO(n)$ is rationally homotopy cocommutative.

Is this correct? If not, where's my mistake? If yes, how to interpret that fact? The group $SO(n)$ doesn't really "look" commutative at all. Maybe I'm assigning a stronger meaning than I should to the commutativity up to rational homotopy?

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In fact every connected Lie group is rationally homotopy commutative, in the sense that every connected Lie group is rationally homotopy equivalent (as a rational H-space) to a product of odd spheres, which are rationally homotopy commutative. This is an extremely weak form of commutativity, though.

Note that the stable orthogonal and unitary groups are already commutative in the strong sense that they form infinite loop spaces, even though they don't look particularly commutative either. So, loosely speaking, rational homotopy theory has a hard time telling apart the orthogonal and stable orthogonal groups, which means the most important differences are in torsion. (This is easiest to see for the homotopy groups.)

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