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.