Perhaps
Simplify[2A1^2+A1 B2+2B1^2-B2 A1+2C1^2+C1 D2+2D1^2-D2 C1-
2(A1^2+B1^2+C1^2+D1^2)+2h1^2-
(A1 B2-B2 A1+C1 D2-D2 C1)+h12]
which returns
2*h1^2 + h12
but this is perhaps not because of the reason you think because
A1 B2-B2 A1+C1 D2-D2 C1==0
unless you might be thinking and have not told Mathematica that these are noncommutative products
Ah, duplicate
here
with more methods than I provided.
