Dimension formula for an algebraic group homomorphism ($\dim G=\dim \ker \phi +\dim \mathrm{im}\phi$)

The kernel is a closed identity fiber. The image is closed because a constructible subgroup is closed. Every nonempty fiber is a kernel translate, so the fiber dimension theorem gives $\dim G=\dim \ker \phi +\dim \mathrm{im}\phi$, even for inseparable morphisms.