The verification principle for $\lambda$-rings says (if I'm understanding correctly) that if you have a $\lambda$-ring $A$, and an equation using only $\lambda$-ring operations (addition, multiplication, negation, $0$, $1$, and the $\lambda^i$ -- as well as your variables of course), then the equation is true in $A$ if and only if it is true when the variables are restricted to be sums of one-dimensional elements of $A$. An element $x$ is said to be one-dimensional if $\lambda^i(x)=0$ for all $i>1$.
I seem to keep getting false results when I try to apply this. For instance, let's consider $\Lambda$, the ring of symmetric functions. The only one-dimensional elements I have been able to find are $0$ and $1$; and while I haven't done anything that would rule out others, they certainly seem to be hard to find. If these truly were the only one-dimensional element, that would make sums of one-dimensional elements just whole numbers. But there are plenty of identities that are true for whole numbers that are not true for all elements of $\Lambda$; for instance, $2\lambda^2(x)=x^2-x$ is true for whole numbers but false when $x=e_1$.
Presumably, there are some one-dimensional elements that I'm missing here. What are they? What are the one-dimensional elements in $\Lambda$? Or am I making some other error instead?
Edit: It's the latter, as Darij Grinberg points out, but an answer to the original question -- presumably a proof that $0$ and $1$ are the only $1$-dimensional elements -- would still be nice, so I'll leave the question open.