The grammar of the parenthesis clause

Among the following sentences, which is right?

• John, when work alone, is very productive.
• John, when working alone, is very productive.

And also, what is this kind of clause called?

The first one doesn't seem to be correct to me, unless you write it as:

John, when he works alone, is very productive.

John, when working alone, is very productive.

The second is correct.

I'd say they are both Subordinate Adverbial Clauses, which are introduced by adverbs such as when, before, after, until, once, as soon as, etc.

EDIT: There are many types of Adverbial Clauses, this one is "Adverbial Clause of Time".

Subordinating conjunctions, such as "when", and "though" can take either:

• a full finite clause (with a subject and finite verb, as "when he works alone"), or
• an adjectival phrase, where the adjective may be a participle, as "when working alone" or "when seen by somebody else", but can also be an ordinary adjective, as "when ready".

They cannot take a non-finite clause with an infinitive verb, as "when work alone".

Several terms exist to describe a clause like when working alone. The most satisfying one I know is elliptical clause: a clause in which words are omitted that the reader can easily infer. In this case, a form of the verb to be has been omitted, as well as a pronoun: when [he is] working alone. The same construction is possible with other conjunctions:

Her father, though [he was] seriously hurt by his fall, kept his face calm and his voice low.

This construction also exists with adjectives instead of participles:

The lid of the container, though [it was] hotter than before, was not yet melting.

I think the traditional term elliptical clause is the most elegant way to explain how some subordinate conjunctions can seemingly do without finite verbs; the simplest model to describe a clause is that it requires a finite verb, be it explicit or implicit. I am not a big fan of grammar books that use the term clause so loosely as to include anything that can have arguments/complements/modifiers; there are many words that can have arguments, such as participles:

The Emperor feared that Bavaria, teeming, as always, with dissenting nobles and fanatical priests, might vote for the Archbishop of Cologne next time.

Some would call teeming, as always, with dissenting nobles and fanatical priests a clause. But what will they say if I substitute rife?

The Emperor feared that Bavaria, rife, as always, with dissenting nobles and fanatical priests, might vote for the Archbishop of Cologne next time.

Would they call this a clause as well? They should, because both sentences are nearly they same. But calling an adjective with a few arguments a clause seems like a very bad idea. We could call anything that can have arguments a clause, then; and the adjective should still be called a "clause" even when it doesn't have any of those: for why should the presence of an argument change the lexical category of a word, if the word itself still functions in precisely the same way?

Let's just stick with the simplest model possible and call the teeming example a participial phrase; we could add that it is used attributively, or perhaps predicatively (though I am not a fan of this vague term). A participle is not a clause unless it is clear that a finite verb has been omitted. Certainly, in the end it doesn't matter which terms and models we choose, as linguistic descriptions are not Dinge an sich; but this is the simplest choice, if we don't want to mess up many other terms too.