On page $12$ of the second edition of John McCleary's "A User's Guide to Spectral Sequences," he gives the following example.
Suppose the $E_2$ of a spectral sequence is $\Bbb Q[x,y,z]/(x^2,y^4,z^2)$, where the bi-degrees of $x,y$, and $z$ are $(7,1)$, $(3,0)$, and $(0,2)$ respectively. Further, suppose that $d_2(x)=y^3$ and $d_3(z)=y$.
The entire $E_2$ page can then be written out since it only has $16$ terms:
$$\begin{array}{c|cc} 4 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ 3 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & zx & \cdot & \cdot & zxy & \cdot & \cdot & zxy^2 & \cdot & \cdot & zxy^3\\ 2 & z & \cdot & \cdot & zy& \cdot & \cdot & zy^2 & \cdot & \cdot & zy^3 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ 1 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & x & \cdot & \cdot & xy & \cdot & \cdot & xy^2 & \cdot & \cdot & xy^3\\ 0 & 1 & \cdot & \cdot & y & \cdot & \cdot & y^2 & \cdot & \cdot & y^3 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \hline & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 &16 \end{array}$$
By assumption, $d_2(x)=y^3$, and by the Leibniz rule, $d_2(zx)=zy^3$. Next, $d_3(z)=y$, and thus $d_3(zy)=y^2$. Similarly, $d_3(zxy)=xy^2$ and $d_3(zxy^2)=xy^3$.
Conclusion: all other differentials are $0$, so the sequence stabilizes on $E_4$, shown below.
$$\begin{array}{c|cc} 4 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ 3 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & zxy^3 & \cdot\\ 2 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & zy^2 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ 1 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & xy & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ 0 & 1 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \hline & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 \end{array}$$
Here is my issue: the Leibniz rule here is that, for $x\in E_k^{p,q}$ and $y\in E_k^{p',q'}$, $$d_k(xy)=d_k(x)y+(-1)^{p+q}xd_k(y).$$ In this case, we have $$d_3(zy)=d_3(z)y+(-1)^{0+2}zd_3(y)=y^2,$$ but also $$d_3(yz)=d_3(y)z+(-1)^{0+3}yd_3(z)=-y^2.$$ Since $E_2$ is a commutative $\Bbb Q$-algebra, this forces $y^2=0$. This seems to be a contradiction, i.e., there can be no such spectral sequence with the prescribed differentials.
Either I am fundamentally misunderstanding some convention, or the example needs to be amended. As you might suspect, I am new to spectral sequences, so I wanted to clear this up, and hopefully also help confused future readers.
There also appears to be a related error later in the book. On page $15$, McClearly says that the Poincaré series for the $E_2$ page would be $$P(E_2,t)=(1+t^{11})(1+t^4+t^8+t^{12})(1+t^3).$$ As users noted on MathOverflow, the correct series is $$P(E_2,t)=(1+t^8)(1+t^3+t^6+t^9)(1+t^2).$$ One person suggested that a previous version of this example may have used the bidegrees $|x|=(10,1)$, $|y|=(4,0)$, and $|z|=(0,3)$. If so, then we do not run into the same error $y^2=0$. However, we incur a similar issue on the second page since $d_2(zx)=(-1)^{0+3}zy^3=-y^3z$ and $d_2(xz)=y^3z$.