Does a spherical building embed in a building of type $A_n$? 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?
 A: If $B$ is of sufficiently high rank (at least 5) then it comes from a polar space, something that is well-known. If the singular subspaces are Desarguesian then it's known to be embeddable in the projective space in the usual way. See Chapter 8 of the book "Diagram geometry" by Buekenhout and Cohen. So in this case (as well as in slightly more general case of $B$ coming from a polar space with Desarguesian singular planes) the answer is yes.
I don't know about the non-Desarguesian case (as well as about small rank non-polar space case, i.e.  $B=F_4$, or $B=E_k$ for $k=6,7,8$, case), perhaps there is more in loc.cit.
A: A comment on the metric aspect of the question.
In the case of the embedding of a polar space $X$ into a projective space $\tilde{X}$ described by Buekenhout and Cohen, hyperplanes in $X$ are sent to points in $\tilde{X}$, and points in $X$ are sent to subspaces in $\tilde{X}$ which are not hyperplanes (unless $X$ is already a projective space). In building terms, this translates to a (poset) embedding of $B$ into $\tilde{B}$ of type $A_n$, whose image contains vertices of type $1$, but no vertex of type $n$.
This cannot be an isometric embedding with respect to the CAT(1) metrics (even after rescaling), since opposite points in $B$ should be sent to opposite points in $\tilde{B}$, because of the non-uniqueness of geodesic segments between them. But the points in $\tilde{B}$ opposite to a vertex of type $1$ are vertices of type $n$.
