When did Odin and Loki became blood brothers?

Question

In the beginning of the Lokasenna, Bragi denies Loki a seat with the gods. However, Loki reminds Odin that the two of them had their "blood mixed", and that Odin had promised that neither of them would drink with the gods unless both of them were welcomed. Odin is then left with little choice but to order Vidar to find a seat for Loki:

Bragi spake:

8. "A place and a seat | will the gods prepare
No more in their midst for thee;
For the gods know well | what men they wish
To find at their mighty feasts."

Loki spake:

9. "Remember, Othin, | in olden days
That we both our blood have mixed;
Then didst thou promise | no ale to pour,
Unless it were brought for us both."

Othin spake:

10. "Stand forth then, Vithar, | and let the wolf's father
Find a seat at our feast;
Lest evil should Loki | speak aloud
Here within Ægir's hall."

_{Source: Poetic Edda/Lokasenna, Wikisource.}

Is there a story that explains how Odin and Loki came to be blood brothers?

Answer 1

According to an annotation by translator Henry Adams Bellows,

There exists no account of any incident in which Othin and Loki thus swore blood-brotherhood, but they were so often allied in enterprises that the idea is wholly reasonable. The common process of "mingling blood" was carried out quite literally, and the promise of -which Loki speaks is characteristic of those which, in the sagas, often accompanied the ceremony; cf. Brot af Sigurtharkvithu. (http://www.sacred-texts.com/neu/poe/poe10.htm#page_151)

Assuming this information is correct, there is no extant story for this.

Answer 2

In the first chapter of “Demonology and Devil-lore”, Moncure Conway proposes that Loki's reminder to Odin in the Lokasenna suggests the two gods may have originated from the same primitive concept; that they once were the good and evil side of the same coin:

The intermediate processes by which the good and evil were detached, and advanced to separate personification, cannot always be traced, but the indications of their work are in most cases sufficiently clear.

...

Not less clear is the illustration supplied by Scandinavian mythology. In Sæmund’s Edda the evil-minded Loki says:—

Odin! dost thou remember
When we in early days
Blended our blood together?

The two became detached very slowly; for their separation implied the crumbling away of a great religion, and its distribution into new forms; and a religion requires, relatively, as long to decay as it does to grow, as we who live under a crumbling religion have good reason to know.