interact. A little thought shows us:
So, when we compse shears with diagonals, we are left with "twisted shears".
The "main objects" are the shears (which are normal), and the "twists" are
provided by the diagonal.
The intuition for why the twisted obect (shears) should be normal
is that the twisting (by conjugation) should continue to give us twisted
objects (shears). The "only way" this can resonably happen is if the twisted
subgroup is normal: ie, invariant under all twistings/conjugations.
§ How the semidirect product forms
From the above computations, we can see that it is the shear transform that
are normal in the collection of matrices we started out with, since
. Intuitively, this tells us that it is the diagonal
part of the transform composes normally, and the shear part of the transform is
"twisted" by the diagonal/scaling part. This is why composing a shear
with a diagonal (in either order --- shear followed by diagonal or vice versa)
leaves us with a twisted shear. This should give a visceral sense of "direct
product with a twist".
§ Where to go from here
In some sense, one can view
all semidirect products as notationally the same as this example
so this example provides good intuition for the general case.