I observed two slightly different following definitions of the $n$th group of Milnor's K-theory of a field $k$.
The first one is $$K^M_n(k)=(k^\times)^{\otimes n}/G_1=\langle\{a_1\otimes\dots\otimes a_n:\exists 1\leq i\leq n-1, a_i+a_{i+1}=1\}\rangle$$ This one is used in Milnor's Algebraic K-theory and quadratic forms.
The second one is $$K^M_n(k)=(k^\times)^{\otimes n}/G_2=\langle\{a_1\otimes\dots\otimes a_n:\exists 1\leq i<j\leq n, a_i+a_j=1\}\rangle$$ This one is used in Gille and Szamuelly's Central Simple Algebra and Galois Cohomology.
The two quotients respect the tensor product, i.e. one can define a ring-product with both of these.
The class of $a\otimes b\otimes(1-a)$ is equal to zero in $K^M_3(k)$ by the second definition, but it is not clear whether or not it is equal to zero by the first one.
My question is, are these two definitions equivalent? In other words, is the inclusion $G_1\subset G_2$ an equality?
There is even a third definition, which defines the Milnor's K-ring directly from the tensor product ring $Tk=\bigoplus_n(k^\times)^{\otimes n}$ $$K^M(k)=Tk/(x\otimes(1-x))$$ and $K^M_n$ is defined to be the $n$-level of this graded ring.
Is is third definition equivalent to one of the two first definitions?
In fact the two definitions are equivalent. One can prove that $a\otimes(-a)\in G_1$, and consequently $a\otimes b+b\otimes a\in G_1$, and hence $G_1=G_2$. This is done in Milnor's book.