Mathfest 2018 Puzzles: Quaternion Groupdoku

Without going at all into application or importance, I'm going to start off giving just enough information about the quaternion group to solve this puzzle, even if you've never heard of a group or the quaternions before.

We have eight elements, 1, -1, i, -i, j, -j, k, and -k, which we are filling into the cells of a grid.  The clue in each cage represents the product of the entries in those cells, computed from left to right or top to bottom.  We just need to understand how to multiply.

A full multiplication table is given here at Groupprops and here at Mathworld.  There's also a table at wikipedia, though their notation uses e for our 1 and bars for our negatives.  The table can also be summed up pretty quickly in words.  Negatives can be pulled out and behave as expected, with two negatives cancelling each other out.  If we square 1 or -1, we get 1.  If we square any other element we get -1.  Finally ij=k, jk=i, and ki=j, and if we reverse the order of those elements we get the negative of the same element as our product.

Link to High Quality PDF for Printing.

I love making puzzles over finite groups like this for a number of reasons.  While solving puzzles over such groups, certain rules inevitably become apparent.  Properties of the group start to reveal themselves, without having to be stated as theorems.  They can still be proven as exercises, but these facts are naturally discovered instead of given at the back of a textbook section.  Plus any additional reason for someone to discover a group for the first time, or to learn more about what goes on in a group they know is always a wonderful thing.