Imperative: how to distinguish independent clauses and a compound predicate?

Question

Consider the following imperative-sentence structure

Do 𝑋 and do 𝑌.

(Source: https://www.chicagomanualofstyle.org/qanda/data/faq/topics/Commas/faq0067.html ) where I believe (but do not know this for sure) that the two “do”s are two placeholders for two occurrences of one or two verbs.

Example from mathematics:

Note that 𝑚 < 𝐿 and apply (*).

Here, (*) is the reference to another formula.

This sentence can be viewed as two independent clauses connected by a conjunction or as a single clause with a compound predicate, cannot it? Why?

Another example in the same vein:

Note that 𝐿 ⩽ [ … a long chain of inequalities occupying slightly more than one line … ] < (𝐺−1)𝐿 and apply (*).

How about this one? Do we have a compound predicate or two independent clauses? Why?

For each of the two exemplary sentences: is a comma before “and”

1. necessary,

2. forbidden,

3. optional with a change in the meaning, or

4. optional without a change in the meaning?

PS. Note that we do not refer to poetry or even high-level prose here (where the writers sometimes take their liberty to intentionally abuse the language to make a point) but rather to daily usage in business, technical documentation, sciences, newspapers and journals (naturally, first class and no yellow press).

Answer 1

The point you seem to miss is whether you analyse this as "two clauses with a coordinator" or "two clauses in a compound predicate". The meaning is exactly the same. There are two things you are commanded to do.

The point of compound predicates is that you can rephrase

John went to the shops and John ate a sandwich.

as

John went to the shops and ate a sandwich.

with no change in meaning

So in the imperative you can rephrase

Go to the shops and eat a sandwich.

as

Go to the shops and eat a sandwich.

with no change in meaning.

But in the imperative there is no change in form and no change in meaning. So it totally doesn't matter.

Commas are optional, with no change in meaning.

Answer 2

Yes, each of your example sentences can be interpreted either as one clause (containing one subject and two predicates) or as two clauses (each containing one subject and one predicate). I'll demonstrate with your first sentence:

[[You]] do 𝑋 and do 𝑌.
[[You]] do 𝑋 and [[you]] do 𝑌.

(The same can be done with your other sentences, even if the two verbs are different.)

In general, a comma must be included whenever a FANBOYS conjunction connects two main clauses, but the comma is often omitted with short clauses. How short? There is no fixed rule, and this issue has been discussed on SE, for example here: https://english.stackexchange.com/questions/119839/when-to-use-a-comma-before-and. It's therefore impossible to state definitively which of your four options is correct, but let me give some examples:

Necessary: "Girls proceed directly to the left by the gymnasium, and boys walk to the right by the cafeteria." (However, some people might consider a comma optional even in this case.)
Forbidden: "Oh, well. Live and learn." (I don't think that any reasonable person would insert a comma before "and".)
Optional with a change in the meaning: "Walk and feed the dog." vs. "Walk, and feed the dog."
Optional without a change in the meaning: "Walk around the block and feed the dog." vs. "Walk around the block, and feed the dog."