I don’t think you’re right

You don’t think Maths textbooks are right??

The wiki page

is full of disinformation. Note that they literally never cite any Maths textbooks

as an example of “elementary arithmetic.”

And whichever Joe Blow My Next Door Neighbour wrote that is wrong

as an example in “elementary algebra.”

Algebra isn’t taught until high school

That implies that yes, this is arithmetic,

No, anything with a(b+c) is Algebra, taught in Year 7

the introduction of variables is what makes it algebra

and the rules of Algebra, which includes a(b+c)=(ab+ac). There is no such rule in Arithmetic.

It doesn’t matter what course finally teaches it to you

It does if you’re going to argue over whether it’s Arithmetic or Algebra.

not by definition part of that domain

The Distributive Law is 100% part of Algebra. It’s one of the very first things taught (right after pronumerals and substitution).

It’s been ages since I took it

I teach it. We teach it to Year 7, at the start of Algebra