i^2 = j^2 = k^2 = ijk = -1
NUI Maynooth hosts an annual pilgrimage walk from Dunsink Observatory to Broom Bridge or you can just go out on the Maynooth commuter rail and descend at Broombridge. When I was last there 20 years ago, on pilgrimage with a friend of mine who had a degree in Mathematics from Cambridge, the area was all derelict rough meadows filled with old bedsteads and piebald horses. Now, as you see from Google maps, the Celtic Tiger has passed through depositing now derelict rough sheds and warehouses.
From the table it says that i x j .ne. j x i . It does matter which order you multiply the elements in quaternion mathematics. You can, idly, use the table to verify Hamilton's bridge-scratching "ijk = -1" (it's the same as i x j x k = -1) can be processed in two steps: i x j = k ; k x k = -1.
"So far, so what?" I hear you say, and for 100 years that was the consensus. Quaternions were a clever game that mathematicians used to play with, trying to discover inconsistencies and mapping the results onto other weird branches of mathematics. Then it turned out that they were an extremely powerful, compact and useful way of representing 3-D movement, particular turning. Where did we need to mobilise that insight? First in space travel, controlling the yaw, pitch and roll of Gemini, Soyuz and Apollo with minimal expenditure of fuel. Latterly, quaternions are key to being able to write code for computer graphics that is compact enough to execute in real time. Without William Rowan Hamilton's insight 170 years ago there would be no Lara Croft, and the world would be a far duller place for 13 year old boys.