Elbow (or ELBOW) is an organization or concept that exists in the world of Kill the Past. They are mentioned by name only a small handful of times in the series, and are usually referred to only obliquely if at all. Elbow seems to be a secret organization that several other factions are merely divisions of, and is implied to have a wide-reaching influence. However, most people are unaware of this. Elbow is linked to multiple education-based conspiracies key to the plots of the games, making Elbow the closest thing the series has to an overarching antagonist.
Elbow is first mentioned by Morichika Nakategawa in case#5:lifecut of The Silver Case as the codename for the Ayame Maspro of the Shelter Kids Policy. Although this use of the name seems random, it is given in the context of being some sort of explanation as to what Nakategawa is saying about factions to Ryu Munakata, a topic that elucidates what Elbow actually seems to be. According to Naka, the boundaries between the factions of the TRO, CCO and FSO were nonexistent, and everyone was actually part of one big group. Although it is not explicitly stated, it can be presumed that they are talking about Elbow. This suggests that the Shelter Kids Policy, and the factional rivalries involved with it, were planned by Elbow.
In Flower, Sun, and Rain, Elbow is much more explicitly linked to the plot. Although they are only mentioned by name in Request 18: An American in Paris, they are directly stated to be the ones behind the breeding of hyenas for silver eyes on Lospass Island. Edo Macalister is said to be a high-level Elbow administrator, but in the course of the game, he is attempting to betray them after discovering their secrets. Edo describes them as "the brainwashing facility of an advanced nation". It is also possible that they are involved in the creation of the Eleven Children of Eleki Island, as they seem to be the organization that the Heinous Crimes Unit is going after for the "Euro Maspro", but this is not clarified.
They are also mentioned in the game's ending text, which describes them as the "Summit of Advanced Nations" and "comprised of secret organizations from around the world". It claims that Elbow used Tokio Morishima's hyena to 'regenerate' the "Next Maspro" before immediately ceasing operations, and also says that they concealed the existence of Lospass. What any of this means is unknown, as Tokio's hyena and the Next Maspro have not been mentioned since.
Elbow is also mentioned in killer7, but only in the Japanese version of the game, in which "ELBOW" is Emir Parkreiner's "national class", given in place of his social security code. (In other versions of the game, Emir's code is "EAGLE".) This would seem to suggest that Emir is an Elbow project; it is possible that Coburn Elementary School itself was at one point, given its similarities to the Shelter Kids Policy. This would also explain why Elbow administrator Edo recognizes Garcian Smith at the Union Hotel.
Another possible reference to Elbow in killer7 is the International Ethics Committee, which operates around the globe and calls its agents "Hyenas".
There are various references in the series to there always being someone higher calling the shots, of everything being according to someone unknown's plan, and in killer7 of history being explicitly pre-determined. This is a recurring theme in the series' absurdist worldview, which Elbow can be seen as the ultimate expression of: an organization always one step bigger and more all-encompassing than the last.