case: (a,b,c) = (x,w,q)

a     a
  b b

The variant with (a,b,c) = (x,w,q) occurs as the mere base-edge alternated faceting s2s4x obtained from the op.

Incidence matrix according to Dynkin symbol

ab ba&#c   → height = sqrt[c2-(b-a)2/2]
             a < b
             c > (b-a)/sqrt(2)

o. o.    | 4 * | 1 1 1 0 0 | 1 1 1 0
.o .o    | * 4 | 0 0 1 1 1 | 0 1 1 1
---------+-----+-----------+--------
a. ..    | 2 0 | 2 * * * * | 1 1 0 0
.. b.    | 2 0 | * 2 * * * | 1 0 1 0
oo oo&#c | 1 1 | * * 4 * * | 0 1 1 0
.b ..    | 0 2 | * * * 2 * | 0 1 0 1
.. .a    | 0 2 | * * * * 2 | 0 0 1 1
---------+-----+-----------+--------
a. b.    | 4 0 | 2 2 0 0 0 | 1 * * *  rectangle
ab ..&#c | 2 2 | 1 0 2 1 0 | * 2 * *  trapezium
.. ba&#c | 2 2 | 0 1 2 0 1 | * * 2 *  trapezium
.b .a    | 0 4 | 0 0 0 2 2 | * * * 1  rectangle

or
o. o.    & | 8 | 1 1 1 | 1 2
-----------+---+-------+----
a. ..    & | 2 | 4 * * | 1 1
.. b.    & | 2 | * 4 * | 1 1
oo oo&#c   | 2 | * * 4 | 0 2
-----------+---+-------+----
a. b.    & | 4 | 2 2 0 | 2 *  rectangle
ab ..&#c   | 4 | 1 1 2 | * 4  trapezium