Bourbaki's Algebra (1998), Ch. II §1.7, introduces the following notation for a specific type of direct sum:
When all the $E_i$ are equal to the same $A$-module $E$, the direct sum $\bigoplus_{i\in I} E_i$ is also denoted $E^{(I)}$: its elements are the mappings of $I$ into $E$ with finite support.
Now, I understand the motivation behind this notation: $E^I = \{ f : I \to E \}$, so $$E^{(I)} \stackrel{\text{def}}{=} \{ f:I\to E \mid \operatorname{supp}(f)<\infty \}$$ seems reasonable (in fact, if $I$ is finite, the two sets are of course equal to each other).
Bourbaki also uses this notation to then write e.g. $$E^{(A\times B)} = \text{set of formal (finite) linear combinations of elements }(a,b).$$
The question: I have the feeling this $E^{(I)}$ notation might conflict with other notations, or also that maybe no one besides Bourbaki uses it anymore.
Some quick advice (authors that use it, other uses of the notation, personal recommendations, etc.) would be much appreciated. Thanks!
There are some people who use it often, and some people who wouldn't know what that means. I use this notation myself, but I usually (depending on audience) also say what it means the first time I use it.
If you want something almost universally understood without needing any explanation, I think $E^{\oplus I}$ or $\bigoplus_I E$ is a good choice.