Problem

I get a work e-mail with an calendar invitation and accept it. It is stored in my private calendar instead of the work calendar.

Solution

Calendars in Thunderbird's lightning allow the selection of an e-mail address. Invitations to that e-mail address are stored in that calendar.

I used the "provider for google calendar" addon for thunderbird: https://addons.thunderbird.net/de/thunderbird/addon/provider-for-google-calendar/. It does not support selecting the e-mail address.

But google has proper caldav support, so you do not need that plugin anymore.

Next problem: Lightning only supports calendar