The so-called "double covering" refers to the idea that this gluing-together of polar opposites can be undone. This can be explained relatively simply, although it does require the introduction of some mathematical notation. The first step is to blurt out "Lie algebra". This is a vector space endowed with the property that two vectors can be multiplied. This arises because a tiny rotation about the x-axis followed by a tiny rotation about the y-axis is not the same as reversing the order of these two; they are different, and the difference is a tiny rotation in along the z-axis. Formally, this inequivalence can be written as , keeping in mind that x, y and z are not numbers but infinitesimal rotations. They don't commute.
The so-called "double covering" refers to the idea that this gluing-together of polar opposites can be undone. This can be explained relatively simply, although it does require the introduction of some mathematical notation. The first step is to blurt out "Lie algebra". This is a vector space endowed with the property that two vectors can be multiplied. This arises because a tiny rotation about the x-axis followed by a tiny rotation about the y-axis is not the same as reversing the order of these two; they are different, and the difference is a tiny rotation in along the z-axis. Formally, this inequivalence can be written as , keeping in mind that x, y and z are not numbers but infinitesimal rotations. They don't commute.