A two-component-two-dimensional coupled with one-component-three-dimensional (2C2Dcw1C3D) flow may also be called a real Schur flow (RSF), as its velocity gradient is uniformly of real Schur form. The thermodynamic and ‘vortic’ fine structures of 2C2Dcw1C3D flows are exposed and, in particular, the Lie invariances of the decomposed vorticity 2-forms of RSFs in d-dimensional Euclidean space E d for any interger d ≥ 3 are also proved. The two Helmholtz theorems of the complementary components of vorticity found recently in 3-space RSF is not coincidental, but underlied by a gen- eral decomposition theorem, thus essential. Many Lie-invariant fine results, such as those of the combinations of the entropic and vortic quantities, including the invariances of the decomposed Ertel potential vorticit 3-formsy (and their multiplications by any interger powers of entropy), then follow.