Here is a distributive law lesson for grade 4

That's the Distributive Property actually. The dead giveaway is the multiply sign, as in "The Distributive Property of Multiplication over Addition". There's no Multiply sign in The Distributive Law, a(b+c)=(ab+ac)

Here’s another, and another.

Also The Distributive Property. "The distributive law says that multiplying a number by a group of numbers added together is the same as doing each multiplication separately" - no, the Distributive Property says that.

These were the first results

Welcome to the problem with using the internet and not looking at Maths textbooks

It being used in an algebra course doesn’t mean it’s in the domain of algebra

It being taught in Algebra most certainly does mean it's in the domain of Algebra

Algebra is also used in calculus, but algebra isn’t the domain of calculus, correct?

It's all Algebra. You can't do Calculus if you haven't learnt Algebra yet, just like you can' do a(b+c) if you haven't learnt Algebra yet.

It’s algebra when it’s using variables

and the rules of Algebra, like a(b+c)=(ab+ac). Arithmetic doesn't have any rules that aren't in Algebra, but Algebra does have rules which aren't in Arithmetic.

and you’re solving for an equation

I can solve 1+1= without using Algebra

2(3+4) is arithmetic

Nope, it's Algebra

2(x+4)=0 is algebra

Yep, now substitute x=3 in 2(x+4) and tell me what you get 😂

the application of the operations of addition, subtraction, multiplication, and division to them

Yep. Notice how Distribution was not mentioned?? 😂

and formal manipulations

Yep, such as a(b+c)=(ab+ac)

rather than specific numbers

Soooo, a+b is Algebra, but 2a+3b+4 isn't Algebra, because it has specific numbers in it?? 😂

Note: Algebra includes the use of arithmetic

Yep, it sure does.

t being used in algebra does not mean it is part of algebra

NOT being used in Arithmetic means it's not part of Arithmetic. 🙄 You know we've only had Brackets in Maths for 300 years, and that Arithmetic is much older than that, right?