Let $E$ be the exterior algebra over $\mathbb{F}_2$ on countably infinitely many generators $x_0,x_1,x_2\ldots$ as discussed in Proposition 1.5 of the following paper. $E$ is a commutative unital $\mathbb{F}_2$-algebra, and has a unique $\mathbb{Z}$-grading where each $x_i$ has degree $2^i$.
Let $I$ be an ideal in $E$ contained in the ideal generated by all products of pairs of generators $x_ix_j$. I want to see if I can get a $\mathbb{Z}$-grading on the quotient algebra $E/I$, and it looks like this should work if I can find an isomorphism of algebras $E/I\to E/J$ where $J$ is an ideal generated by products of generators $x_{i_1}x_{i_2}...x_{i_n}$.
One way to do this might be to find an automorphism $\phi$ of $E$ with $\phi(I)=J$. This might involve a change of variables of some sort, but I have no idea how to proceed.
My motivation for looking into this is to find a $\mathbb{Z}$-grading of $E/I$ in order to apply results in the linked paper to test for self-injectivity using blocks and transporters. Any tips much appreciated.