I'm interested in the question in the title:

Does a spherical building $B$ always embeds in a building $\tilde B$ of type $A_n$ for some $n$?

By embedding I mean an isometric embedding with respect to their $CAT(1)$ metric. If the question has a positive answer, then how does this embedding work? For instance, does an automorphism of $B$ extend to an automorphism of $\tilde B$?

Edit: I forgot that there are strange polygons. I would like to restrict the question to the case of irreducible thick buildings of rank at least 3. In particular, they are Moufang.

Edit 2: As Dima Pasechnik pointed out, if the building $B$ has residues of type $A_2$ which are non-Desarguesian projective planes, then we cannot have such an embedding, since projective spaces of dimension $\geq 3$ must be Desarguesian. This is a problem if $B$ is of type $B_3$ or $F_4$. In the former case, it is known that there are non-embeddable buildings (see Tits' "Buildings of spherical type and BN-pairs", chapter 9). Hence the following question:

What about type $F_4$?

Buildings of type $B_n$ with $n\geq 4$ are embeddable (see Tits' "Buildings of spherical type and BN-pairs", chapter 8) into a projective space of possibly infinite rank. So, I assume that the answer to my quesion in this case is "No in general" (because of the infinite rank). Is this correct?

Buildings of type $D_n$ should be embeddable, since they correspond to the group $SO_{2n}(k)$ for some field $k$. Hence these buildings are embeddable in $A_{2n}$.

Type $E_n$, $n=6,7,8$, should not be a problem either, since they correspond to algebraic groups. Is this also correct?