Let be $k$ a field and $G$ the group of the triangular superior matrices in $GL_3(k)$. Prove that $G$ is the semidirect product $U\rtimes{D}$, where $U$ is the set of upper triangular matrices with $1$s on the diagonal and $D$ is the set of diagonal matrices in $GL_3(k)$.
My path: I tried to say that every semidirect product is determined by the choise of homomoprhism $D \rightarrow Aut(U)$. Such homomorphism is determined by its value on a generator of D, but I'm not sure if this is the right way to continue the proof, any ideas?