Among simple Lie groups, $Spin(8)$ is the most symmetrical one in the sense that $Out(Spin(8))$ is the largest possible group. A description of this outer automorphism groups is as follows. $Spin(8)$ has 2 half-spinor representations and one vector representation (coming from standard representation of SO(8)) all of them of dimension 8. In general one can pre-compose a representation of a Lie group with an automorphism of the group to get another representation. In this special case $Out(Spin(8))$ permutes above three representations and this we get an isomorphism of $Out(Spin(8))$ and $S_3$.

My questions:

1- Is there any way to see an automorphism explicitly which interchanges the vector representation and a half-spinor representation? (It is easy to see there exists such an automorphism. But I'd like to have an explicit construction of such an automorphism.)

2- Apparently there is a 27-dimensionla Jordan algebra which has $Aut(Spin(8))$ as a subgroup of its automorphism group. Can anyone explain what is this Jordan algebra and how should I think about $Aut(Spin(8))$ acting on it?