Endomorphism/Trigonalizable/Canonical additive decomposition/Fact

Theorem about the canonical additive decomposition

Let

$\phi :V⟶V$

denote a trigonalizable $K$-endomorphism on the finite-dimensional $K$-vector space $V$.

Then there exists a decomposition

$\phi ={\phi }_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}+{\phi }_{\mathrm{n}\mathrm{i}\mathrm{l}},$

where ${\phi }_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}$ is diagonalizable and ${\phi }_{\mathrm{n}\mathrm{i}\mathrm{l}}$ is nilpotent, and where

${\phi }_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}\circ {\phi }_{\mathrm{n}\mathrm{i}\mathrm{l}}={\phi }_{\mathrm{n}\mathrm{i}\mathrm{l}}\circ {\phi }_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}$

holds.

Proof, Write another proof