  We consider a field $F$ and positive integers $n$, $m$, such that $m$ is not
divisible by $\mathrm{Char}(F)$ and is prime to $n!$. The absolute Galois group
$G_F$ acts on the group $\mathbb{U}_n(\mathbb{Z}/m)$ of all $(n+1)\times(n+1)$
unipotent upper-triangular matrices over $\mathbb{Z}/m$ cyclotomically. Given
$0,1\neq z\in F$ and an arbitrary list $w$ of $n$ Kummer elements $(z)_F$,
$(1-z)_F$ in $H^1(G_F,\mu_m)$, we construct in a canonical way a quotient
$\mathbb{U}_w$ of $\mathbb{U}_n(\mathbb{Z}/m)$ and a cohomology element
$\rho^z$ in $H^1(G_F,\mathbb{U}_w)$ whose projection to the superdiagonal is
the prescribed list. This extends results by Wickelgren, and in the case $n=2$
recovers the Steinberg relation in Galois cohomology, proved by Tate.
