Hilbert reformulated the quadratic reciprocity law of Gauß as a product formula 
$$
\prod_v(a,b)_v=1
$$
for the various local Hilbert symbols.  For each place $v$ of $\bf Q$, the Hilbert symbol $(\ ,\ )_v$ is a *bimultiplicative* map 
$$
{\bf Q}_v^\times\times{\bf Q}_v^\times\to{\bf Z}^\times
$$
so that, by definition, $(a,b)_v=1$ if and only if $a\in {\rm Im\;} N_b$ where $N_b$ denotes the norm map ${\bf Q}_v(\sqrt b)^\times\to{\bf Q}_v^\times$.  An important property of the Hilbert symbol is that
$$
a+b=1\Longrightarrow (a,b)_v=1,
$$
which makes it a Steinberg symbol.  This property in not listed in older books such as Hasse's *Number theory* but it can be found in all modern treatments, such as Serre's *Course in arithmetic* or his *Local fields*, or Milnor's *K-theory*.

I'm curious as to who first noticed that the Hilbert symbol is a Steinberg symbol. Was it Steinberg himself ?  A precise reference will be appreciated.