  Regarding a Dynkin diagram as a specific point-line incidence structure
(where each line has just two points), one can associate with it a Veldkamp
space. Focusing on extended Dynkin diagrams of type $\widetilde{D}_n$, $4 \leq
n \leq 8$, it is shown that the corresponding Veldkamp space always contains a
distinguished copy of the projective space PG$(3,2)$. Proper labelling of the
vertices of the diagram (for $4 \leq n \leq 7$) by particular elements of the
two-qubit Pauli group establishes a bijection between the 15 elements of the
group and the 15 points of the PG$(3,2)$. The bijection is such that the
product of three elements lying on the same line is the identity and one also
readily singles out that particular copy of the symplectic polar space $W(3,2)$
of the PG$(3,2)$ whose lines correspond to triples of mutually commuting
elements of the group; in the latter case, in addition, we arrive at a unique
copy of the Mermin-Peres magic square. In the case of $n=8$, a more natural
labeling is that in terms of elements of the three-qubit Pauli group,
furnishing a bijection between the 63 elements of the group and the 63 points
of PG$(5,2)$, the latter being the maximum projective subspace of the
corresponding Veldkamp space; here, the points of the distinguished PG$(3,2)$
are in a bijection with the elements of a two-qubit subgroup of the three-qubit
Pauli group, yielding a three-qubit version of the Mermin-Peres square.
Moreover, save for $n=4$, each Veldkamp space is also endowed with some
`exceptional' point(s). Interestingly, two such points in the $n=8$ case define
a unique Fano plane whose inherited three-qubit labels feature solely the Pauli
matrix $Y$.
