I am trying to prove that if $K$ is an algebraically closed field, the commutator subgroup of the group $T_n(K)$ of upper triangular matrices is $U_n(K)$, the subgroup of upper triangular matrices with 1's on the diagonal.
I'm trying to prove this by induction. I proved that the commutator of $T_n(K)$ is contained in $U_n(K)$. I proved the case $n=2$ but am having trouble with the induction step. I don't see how to express the terms in the last column of $U_{n+1}(K)$ in the desired form.