I am reading John Baez's paper on the "The Octonions" and it is chalked full of interesting stuff such as:
$$ F_{4} \cong Aut \left( \mathfrak{h}_{3} \left( \mathbb{O} \right) \right) \cong Isom \left( \mathbb{O}P^{2} \right). $$
This would incline me to believe that:
$$ O \left( n + 1 \right) / O \left( 1 \right) \cong Aut \left( \mathfrak{h}_{n} \left( \mathbb{R} \right) \right) \cong Isom \left( \mathbb{R}P^{n} \right) $$
$$ U \left( n + 1 \right) / U \left( 1 \right) \cong Aut \left( \mathfrak{h}_{n} \left( \mathbb{C} \right) \right) \cong Isom \left( \mathbb{C}P^{n} \right) $$
$$ Sp \left( n + 1 \right) / \left \{ \pm 1 \right \} \cong Aut \left( \mathfrak{h}_{n} \left( \mathbb{H} \right) \right) \cong Isom \left( \mathbb{H}P^{n} \right) $$
$$ \text{for} \ n \ge 3 $$
although I do not see this explicitly written in his paper.
Another interesting fact is that the projection operators of rank 1 correspond to points in the projective space while the ones of rank 2 correspond to lines and so on.
My question is can the representation theory of these Jordan algebras be given a geometric realization as a Borel Weil theory on these projective spaces and where can I find a paper that explains how this all works? Is this more or less the content of Jacques Tits theory of Buildings?