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Is there a way to forward (or to get in any other way) the important mail as they arrive (or shortly after that) from the priority inbox to another mail account without forwarding the unimportant mails?

I have tried to create a filter with is:important but I get a warning message that the filter will not apply to incoming mail and sure enough the defined actions don't get triggered if an important mail arrives.

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Same question as the one you asked yesterday, just with different wording: webapps.stackexchange.com/questions/12337/… – Eight Days of Malaise Feb 13 '11 at 18:07
Yes. This one is worded to match better the topic of this stackexchange site. On the previous one I got the feedback that it is better suited for the apple stackexchange site. Also on that site I received the answer that it is more of a GMail issue and I should work that angle so I'm back here. apple.stackexchange.com/q/8250/3566 – Andris Feb 13 '11 at 19:54
I really hope you find a positive answer here. I'd add an answer "there's no way, it depends on google implementing it and it haven't done it so far", but I rather leave this unanswered. ;) – cregox Feb 14 '11 at 16:58
Yes, seems like Google isn't interested in this: the Priority box feature effectively locks users in, in a way. – Alexander Shcheblikin Oct 11 '13 at 19:12
up vote 3 down vote accepted

The only solution that I could find was to access my Gmail account with a 3rd party service like Mail2Web (not free) via IMAP and look for mails in the [Gmail]/Important IMAP folder.

I ended up using Ultimate Notifier because my goal was to get push notifications to my iPhone for important mail. Details on: http://apple.stackexchange.com/q/8250/3566

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The correct filter is in:important

Place that text in the text-box labeled Has the words when creating a filter.

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Unfortunately, it says it won't work with incoming mail: "Filter searches containing "label:", "in:", "is:", or stars criteria (i.e. "has:yellow-star") are not recommended as they will never match incoming mail." – jowie Sep 11 '13 at 20:51

Google says that label filters will never match incoming mail but that's a lie. label:important works fine for me. label:whatever seems to work fine too, if you created another filter that assigns that label. Perhaps what they were trying to say is what they say clearly anyway in the 2nd page of the filter's definition: the filter works on incoming mail only; mails already received and marked important later are not forwarded.

PS However, if you have another filter that clears the important flag of incoming mail, those mails are still matched by label:important (regardless of which rule is defined first). I only tested label:whatever superficially, not enough to certify it actually works fine.

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I actually had the exact same issue. I wanted to forward all of my important emails to SMS. You could do the same thing in IFTTT, but send it to another email address instead of SMS.

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Typing in category:primary in the Has the words box worked perfectly for me.

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