# Matrix groups transitive on the sphere

I would like a reference with the classification of the subgroups of SO($d$) which are transitive in the unit sphere of $\mathbb{R}^d$ when acting linearly (i.e. as matrices from SO($d$) act on vectors from $\mathbb{R}^d$). Onishchik gives the (abstract) Lie algebras of these groups, without specifying which representations.

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By a celebrated result of Jim Simons, the Lie subgroups of $SO(d)$ acting transitively on the unit sphere in $\mathbb{R}^d$ coincide with Marcel Berger's original list of possible holonomy groups of complete, simply-connected, non-symmetric irreducible riemannian manifolds. The section Berger's classification in the wikipedia entry on Holonomy contains the list. They are the known holonomy groups: $SO(d)$, $U(d/2)$, $SU(d/2)$, $Sp(d/4)$, $Sp(d/4)\cdot Sp(1)$, $G_2 \subset SO(7)$, $Spin(7) \subset SO(8)$, as well as two cases which were since then ruled out: $Spin(9) \subset SO(16)$ and $Sp(d/4)\cdot U(1)$.

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For instance, have a look at the book "Einstein manifolds" by Arthur L. Besse, on page 179. The action of $G_2$ on $S^6$ comes from its lowest dimension nontrivial representation, which is $7$-dimensional: the inclusion $G_2\subset SO(7)$ is realized by recalling that $G_2$ is the group of automorphisms of the real division algebra of Cayley numbers (octonions). The actions of $Spin(7)$ and $Spin(9)$ come from their spin representations. The other groups are classical and act by their standard (natural, vector) representations.

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