The object the author is actually interested in is known as a geometric algebra. One often sees it discussed as an alternative theory for computer graphics or physics as it works well for expressing things like rotations.
I think it is probably not so helpful to merely think of it like a division algebra, and it is better to stay focused on the geometry. Curiously I find it easier to relate “actual” linear algebra to geometry than the thing people often call “linear algebra” that involves writing columns or rows or grids of numbers and manipulating them.
See here: https://en.m.wikipedia.org/wiki/Geometric_algebra
I think it is probably not so helpful to merely think of it like a division algebra, and it is better to stay focused on the geometry. Curiously I find it easier to relate “actual” linear algebra to geometry than the thing people often call “linear algebra” that involves writing columns or rows or grids of numbers and manipulating them.