Insertions
Insertions are a fewest move technique that allows for efficient solving of Skeletons. This is made possible by solving the remaining pieces using a special algorithm at any point in the solve and not necessarily in the end, which can lead to cases that can be solved in fewer moves, sometimes even cancelling moves with the skeleton. Therefore, the use of insertions usually tends to be more efficient than solving the skeleton optimally after its last move.
Contents
History
Insertions are a very old. They were most likely invented by one of the early pioneers of Fewest Moves, Per Kristen Fredlund. In 2020, thanks to research by James Straughan, this was confirmed by Fredlund himself:
“To the best of my knowledge I did indeed come up with the concept of using insertions for fewest moves. I used that (in most cases) for inserting corner 3cycles into skeletal solutions, with 3 or more unsolved corners. I found out that using insertions in that skeletal solution was almost always better than appending the cycles(s) at the end.”
—Per Kristen Fredlund
Naming conventions
Insertions are usually named after the Skeleton they solve. If, for example, the skeleton is a 3c skeleton, the insertion is called a 3c insertion. For more information on how to obtain the names for any given skeleton, refer to Skeleton#Naming conventions.
Which algorithms to use in insertions
Insertions are most effective when short and flexible algorithms are used. This means that algorithms used for insertions should be efficient, easy to set up to or common to encounter and should have different varieties. One example of such an algorithm is R' L F2 R L' U2, which performs a 3cycle. With only six moves, it is very short. The algorithm is easy to set up to, mainly because only three pieces are affected, and has different varieties like R' F2 R L' U2 L, L F2 L' R U2 R' and F2 R L' U2 R' L which can be used to cancel more moves in certain cases.
Commutators also tend to be very good for insertions as they require relatively few moves (minimum of 8 HTM for standard corner and block commutators, 10 HTM for standard edge commutators) and can be generated by the solver at any given point in the skeleton. Thus, they are not confined to one specific case and skeletons where only commutators with setup moves can be inserted are extraordinarily rare.
Movecounts for specific insertion types
Type of insertion  Moves 

Corner 3cycle  5/6 
Edge 3cycle  7 
2 Twisted Corners (2 comms)  8 
3 Twisted Corners (2 comms)  9 
4 Corners (double swap)  9/10 
4 Corners (3 + 1 twisted)  10 
Corner 5cycle  10/11 
2 Corners and 2 Edges (double 2cycle)  10 
The movecounts are taken from Sebastiano Tronto's FMC tutorial, v3, page 33.
Useful insertion algorithms

Insertion Finder
Insertion Finder is a tool that performs a variety of different insertions optimally. This can be helpful to check whether the solver could've done better and is mostly used to practice finding efficient or even optimal insertions.