When the Subject of a non-finite subordinate clause is the same as the Subject of a main clause, we can usually omit the Subject from the subordinate clause.
In English, clauses with a tensed verb must have a Subject, and for this reason, whenever the Subject is missing we will see a non-tensed (non-finite) version of the verb. In addition, any tensed forms of BE will therefore be omitted if the Subject is missing:
- Although she was late, she still attended the meeting.
Although late she still attended the meeting. (Subject and tensed BE omitted)
After I studied for one year, I got 6.5 in my IELTS exam.
- After studying for one year, I got 6.5 in my IELTS exam. (No Subject, verb in non-finite -ing form)
We can of course also omit the Subjects of subordinate clauses which use to-infinitives when they are the same as the Subject in the main clause (because these, like -ing forms, are not tensed):
- For her to pass the exam, Paula will need to get over 60%.
- To pass the exam, Paula will need to get over 60%.
Although it is frowned upon by some style guides, and prescribed against in some exams (for example in SAT exams), in real life people often omit the Subjects of subordinate -ing clauses when they are clearly discernible from the context:
- Without going into details, the party was a complete disaster.
In the sentence above the understood Subject of going is me in other words, the speaker. The Subject of the main clause is, of course, the party. Nonetheless, there is nothing strange or odd about this sentence. In real life, writers should avoid omitting Subjects in subordinate clauses when it will jar the reader or cause confusion. The following is bad writing:
- Lying on the floor bleeding like that, I now wished I hadn't shot him.
The sentence above is bad, because we cannot be entirely confident whether the speaker or the person who was shot is lying on the floor. It doesn't matter that some grammars would allow this if the speaker was on the floor. In real life readers will be confused, because real language users might be referring to either person!
The Original Posters example
could be important.
Although speakers talking casually may omit the Subject of this sentence, it is not a non-finite clause. The omission of Subjects in such sentences is an example of diary drop, which is an unrelated phenomenon.
Modal verbs such as can, could, may, might, shall, should, will, would or must are considered to have tense by most grammarians. In any case they are always considered to head finite clauses. Because of this we cannot freely drop the Subjects of clauses headed by modal verbs.