Rule: Being can introduce an adverbial clause only if it expresses reason or cause.
Examples of errors:
(1) We went home. Being there a half hour, the phone rang.
(2) Being at this point in our discussion, d big problem arises.
(3) Being content with the author's conclusion, I want to say that there--are exceptions.
(4) Being children, we are read fairy tales which tell us about good and evil.
When introducing a (reduced) adverbial clause, being is equivalent to because + subject + be, for example:
Being/Because he was a mechanic himself, he recognized the problem immediately.
His wife, being/because she is a patient woman, puts up with his temper.
The problem, or one problem, in (l)-(4) is that the clauses do not indicate reason or cause, but time ((1), (4)), place ((2)), or concession ((3)). Grammatically, of course, the being construction is acceptable with a causal interpretation, but this does not express the intended meaning:
Because I was there a half hour, the phone rang.
Because we are at this point in our discussion, a big problem arises.
Because I am content with the author's conclusion, I want to say that there are exceptions.
Because we are children, we are read fairy tales
The second problem with reference to (1) and (2) doesn't concern being particularly, but all participial adverbial clauses. This is what is traditionally called the unattached or unrelated participle. The rule is that the understood subject of the non-finite verb (here being) should appear as subject of the main clause (as in the two examples above, and in (3)-(4)). Exceptions include cases where the understood subject appears in a non-subject function in the main clause, e.g.:
Using different words, his statement sounded more polite. (His represents the understood subject of using.)
Using different words, the statement can be made more politely. (The understood subject of using is the unexpressed agent in the main clause--the person who makes the statement.)
Unattached participles are especially common in scientific English (cf. Quirk et al. 1972':§11.45). Being is less likely to be used as an unattached participle than other verbs, however.