Polarization trick

Statement for a biadditive function and a single repeated variable
Suppose $$f$$ is a biadditive function of two variables both from a set $$T$$ and:

$$f(x,x) = 0 \ \forall \ x \in T$$

Then, we have:

$$f(x,y) + f(y,x) = 0 \ \forall \ x,y \in T$$

Note that the converse implication need not be true in general, but it is true if the range of $$f$$ is 2-torsion-free.

This particular case is so important that it has a name. A (biadditive) function satisfying the first condition is termed alternating and a (biadditive) function satisfying the second condition is termed skew-symmetric.

Statement for a multiadditive function and a single repeated variable
Suppose $$f$$ is a function of $$n$$ variables all from a set $$T$$ that is additive in each variable. Suppose we have that:

$$f(x,x,\dots,x) = 0 \ \forall \ x \in T$$

Denote by $$S_n$$ the symmetric group on the set $$\{ 1,2,\dots, n\}$$. We have:

$$\sum_{\sigma \in S_n} f(x_{\sigma(1)},x_{\sigma(2)},\dots,x_{\sigma(n)}) = 0$$

Note that the converse implication need not be true in general, but it is true if the range of $$f$$ has torsion-free threshold at least $$n$$.

General statement
Suppose $$f$$ is a function of $$n$$ variables all from a set $$T$$ that is additive in each variable. Consider a function:

$$\{ 1, 2, \dots, n \} = A_1 \sqcup A_2 \sqcup A_m$$

where the $$A_i$$ are pairwise disjoint nonempty subsets. Note that we can define a function:

$$g: \{ 1,2,\dots, n \} \to \{ 1,2,\dots,m\}$$

that sends a given element $$i$$ to the $$j$$ such that $$x_i \in A_j$$.

Suppose $$f$$ satisfies the identity:

$$f(x_{g(1)},x_{g(2)},\dots,x_{g(n)}) = 0 \ \forall x_1,x_2,\dots,x_m \in T$$

Then, the following is true:

$$\sum_{\sigma \in \prod_{j=1}^m\operatorname{Sym}(A_j)} f(x_{\sigma(1)},x_{\sigma(2)},\dots,x_{\sigma(n)}) = 0 \ \forall x_1,x_2,\dots,x_n \in T$$

Related facts

 * Alternating function condition is transitive
 * Symmetric or skew-symmetric function condition needs to be checked only on a generating set for the symmetric group