It’s necessary that I not be lazy?
Is this sentence correct? Because it sounds odd - but due to the structure of subjunctive:
It’s important /(un)necessary /essential + that + Subject + (should) + V.inf without to.
The sentence
It’s necessary that I not be lazy?
is not a subjunctive form. It is grammatical but a bit unusual, and thus feels awkward. I think this is partly due to the negative form. An equivalent positive form feels better.
- It is necessary that I be diligent?
- It is necessary that I be productive?
Also, when asking this as a question, it is more usual to invert subject and verb:
- Is it necessary that I be diligent?
- Is it necessary that I be productive?
leading to
Is it necessary that I not be lazy?
It is not a very good sentence.
It is structured as a statement, but it has a question mark. Questions should use question syntax.
It has a confusing negative structure, and it uses a subjunctive, in a very odd way.
You would never say this. How about?
Is laziness permitted?
Am I allowed to be lazy?
This sentence means the same, but I can understand this one easily. It took me 5 minutes to work out what your sentence means.
Or you can change the adjective:
Is it necessary to be energetic and industrious?
But all of these questions are odd: Nobody asks permission to be lazy.