[Being] A useful technique, reducing the sentence phrases and pieces into smaller, but logically similar chunks, can make the meaning easier to understand. Being a somewhat thoughtful editor, I would then say it's also a good idea to reorder those chunks - especially for legal Mumbo Jumbo! Keep in mind, for the time being, that doing so will change the meaning. But you can add that back in.
Being that I have the time to write this, let's take your example.
All directors retired in accordance with Article 3 of the Company Ordinance and, being eligible, offered themselves for re-election.
retired in accordance with Article 3 of the
Company Ordinance and, being eligible, offered themselves
for re-election. [Note the comma remains after 'and' !]
Being eligible, all directors offered themselves for re-election.
Being eligible, [A]ll directors offered [.]
themselves for re-election.
So now we have one possible core sentence:
The directors offered. [EDIT: See comment from @John M. Landsberg below]
[Edit 19Sept13: considering Mari-Lou's comment below]
We can also consider the "in accordance ..." as a long phrase, so another reduction could be simply:
The directors retired.
(Then we have to parse down that phrase ! :)
The original sentence was stilted English. Stilted anything is Mumbo Jumbo. Being an easily annoyed editor, I'd say this is much better:
Following Article 3 of the Company Ordinance, all the directors offered themselves for re-election.
Being of good spirits, this aside comment to Strunk and White bashers: guess where I learned this ?