Hi Pierre:
I'm going to refer to the Brightspace LMS as Tool Consumer (TC) and your other LTI-connected service as a Tool Provider (TP) here, for clarity.
LTI 1.x supports "tool launches" from a Tool Consumer to a Tool Provider, as the assumption is that the main user experience happens in the TC and the TP is the one providing the enhanced bit of user experience. This implies that, in the LTI 1.x model, the user is assumed to start in the TC and then call on the TP.
So, can you be a bit more descriptive of your use case? What do you want to achieve here? Is it that you want to support a case where the user starts the user experience in the Tool Provider space, and then SSOs over to the Tool Consumer to do work in the LMS?
Or are you looking to retrieve back a set of API keys for a user right after an LTI launch so that you can, from your Tool Provider, make API calls against the Tool Consumer?
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Viktor