Steinberg symbol

Remarks

Every Steinberg symbol also satisfies $c(a, -a) = 1$ and $c(a, b) \, c(b, a) = 1$ for all $a, b \in F^\times$.

Example: on $F = \R$, let $c(a, b) = -1$ if $a < 0$ and $b < 0$, and $c(a, b) = 1$ otherwise; this is the real Hilbert symbol. It is multiplicative in each variable, since for fixed $b < 0$ it is the sign of $a$, and for fixed $b > 0$ it is constant. For $a \ne 0, 1$ the numbers $a$ and $1 - a$ are never both negative, so clause 3 holds.

Non-example: on $F = \Q$, let $v_2$ be the $2$-adic valuation and $c(a, b) = (-1)^{v_2(a) v_2(b)}$. Clauses 1 and 2 hold because $v_2$ is additive. Clause 3 fails: for $a = 1/2$ also $1 - a = 1/2$, and $c(1/2, 1/2) = (-1)^{(-1)(-1)} = -1$.

The symbols $\set{a, b}$ in $K_2(F)$ form the universal Steinberg symbol, by Matsumoto's theorem.