This may be an elementary question but I hope this question will benefit others as much as myself.
Let $k^4 = Spec \; k[x_1, x_2, x_3, x_4]$.
Writing $Bl_{\mathcal{I}}(k^4)$ as $R = Proj(\oplus_{i\geq 0} I^i)$ where $\mathcal{I}$ is defined by $(x_1-x_2, x_3-x_4)$, what are the degree $1$ elements in $R$?
Are they just $(x_1 - x_2)k[x_1, x_2, x_3, x_4] \oplus (x_3 - x_4)k[x_1, x_2, x_3, x_4]$?
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$\mathbf{Added}:$ By the way, if anyone is interested, I am blowing up a certain diagonal subset in affine $4$-space.