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”
optional with a change in the meaning, or
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).