YES

2 decompositions
 #1 -----------
   2: a() -> b()

 #2 -----------
   3: f(b(),x1) -> g(f(x1,x1))
   4: f(x1,b()) -> g(f(x1,x1))

@Jouannaud and Kirchner's criterion
--- R
   2: a() -> b()

--- S
   2: a() -> b()


@Development Closedness
--- R
   3: f(b(),x1) -> g(f(x1,x1))
   4: f(x1,b()) -> g(f(x1,x1))

--- S
   3: f(b(),x1) -> g(f(x1,x1))
   4: f(x1,b()) -> g(f(x1,x1))


NOTE: input TRS is reduced

original is
   1: f(a(),a()) -> g(f(a(),a()))
   2: a() -> b()
   3: f(b(),x1) -> g(f(x1,x1))
   4: f(x1,b()) -> g(f(x1,x1))

reduced to
   2: a() -> b()
   3: f(b(),x1) -> g(f(x1,x1))
   4: f(x1,b()) -> g(f(x1,x1))