5 Add reference to the problem with Google Hangouts and google takeout solution
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The IMAP interface to Gmail now provides access to chat messages. If you had already set up Gmail IMAP, then you only have to enable the IMAP visibility for the "Chat" label in the Gmail settings (in the "Label" tab, a "show in IMAP" tick box is available for each label).

Update: this method works for Google Chat, but NOT for Hangouts, only conversations of the type Google Chat are exposed via IMAP, roughly all conversations until Spring/Summer 2013, depending on when one has enabled Google Hangouts.

Sources:

The latter link provides an alternative to download via Google Takeout in the JSON format

The IMAP interface to Gmail now provides access to chat messages. If you had already set up Gmail IMAP, then you only have to enable the IMAP visibility for the "Chat" label in the Gmail settings (in the "Label" tab, a "show in IMAP" tick box is available for each label).

The IMAP interface to Gmail now provides access to chat messages. If you had already set up Gmail IMAP, then you only have to enable the IMAP visibility for the "Chat" label in the Gmail settings (in the "Label" tab, a "show in IMAP" tick box is available for each label).

Update: this method works for Google Chat, but NOT for Hangouts, only conversations of the type Google Chat are exposed via IMAP, roughly all conversations until Spring/Summer 2013, depending on when one has enabled Google Hangouts.

Sources:

The latter link provides an alternative to download via Google Takeout in the JSON format

4 add where the option is in the Settings
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The IMAP interface to gmail Gmail now provides access to chat messages. If you had already set up gmail Gmail IMAP, then you only have to enable the IMAP visibility for the "Chat" label in the gmail Gmail settings (in the "Label" tab, a "show in IMAP" tick box is available for each label).

The IMAP interface to gmail now provides access to chat messages. If you had already set up gmail IMAP, then you only have to enable the IMAP visibility for the "Chat" label in the gmail settings (in the "Label" tab, a "show in IMAP" tick box is available for each label).

The IMAP interface to Gmail now provides access to chat messages. If you had already set up Gmail IMAP, then you only have to enable the IMAP visibility for the "Chat" label in the Gmail settings (in the "Label" tab, a "show in IMAP" tick box is available for each label).

3 add where the option is in the Settings
source|link

The IMAP interface to gmail now provides access to chat messages. If you had already set up gmail IMAP, then you only have to enable the IMAP visibility for the "Chat" label in the gmail settings (in the "Label" tab, a "show in IMAP" tick box is available for each label).

The IMAP interface to gmail now provides access to chat messages. If you had already set up gmail IMAP, then you only have to enable the IMAP visibility for the "Chat" label in the gmail settings.

The IMAP interface to gmail now provides access to chat messages. If you had already set up gmail IMAP, then you only have to enable the IMAP visibility for the "Chat" label in the gmail settings (in the "Label" tab, a "show in IMAP" tick box is available for each label).

2 added solution description instead of naked link.
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