I was trying a lazy way to derive rotation matrix, but then I have observed something funny -- there seem to be two kinds of rotations:

Suppose we rotate a book about its vertical axis perpendicular to its plane - the axis is outside of the $\mathbb{R}^2$. However, if we rotate an apple about any axis- the axis is always inside of the $\mathbb{R}^3$.

Another way to see it: For an ant on a book, we can fool it in two ways: one is to rotate the book about an axis perpendicular to the plane; another is to about any axis parallel the book. The Physics laws will be different in two cases, as the ant is in different sorts of non-inertial frames.

Hence, given a $\mathbb{R}^2$ (a subspace of a $\mathbb{R}^3$), there seems to be two kinds of rotations.

Now I wonder if we have these two kinds of rotations in our $\mathbb{R}^3$ space world - somehow rotate things "about the time axis"?

After a few searches on the Internet, I have came across something about hyperbolic rotation(that seems relevant to my question), but sadly, as I know little relativity, I can't understand it.

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    $\begingroup$ Note that if you rotate about an axis parallel to the book, the book does not move in a plane, i.e. it's actually this rotation you need $\mathbb{R}^3$ for, while the rotation about the perpendicular axis happens entirely in the plane $\mathbb{R}^2$ and could be described as such without referring to an axis at all. $\endgroup$ – ACuriousMind Jan 16 '18 at 18:22
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    $\begingroup$ It is a common misconception that rotation happens around the axis. In fact, rotation happens in the plane while the existence of the axis is not guaranteed. There is no axis in 2D. There is one in 3D, but you still can see it as rotation in the plane rather than around the axis. In 4D, you can have two independent rotations at the same time in two orthogonal planes. In this case there would be no invariant axis of rotation in 4D. $\endgroup$ – safesphere Jan 16 '18 at 18:58
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    $\begingroup$ When you say, "rotation around the time axis", you are under the same confusion. Rotation is not around the axis, but in the plane. For example, If your rotation is in the plane XY, then you can see any coordinate that is perpendicular to XY as the "axis of rotation". In this case you have two, Z and T. So rotation around any spatial axis is equivalent to rotation around the time axis. To avoid this confusion, it is better to speak of rotation in the plane. However, if your plane of rotation includes the time coordinate, like XT, then rotation is hyperbolic meaning the Lorentz boost. $\endgroup$ – safesphere Jan 16 '18 at 19:10
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    $\begingroup$ @safesphere thank you so much for the elaborating! it does make more sense to rotate in the plane than about an axis now. $\endgroup$ – Shing Jan 16 '18 at 19:20
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    $\begingroup$ All this lore of @safesphere can be derived by working with the linear transformations of cartesian coordinates that preserve distance and the origin (a representation of rotations). You can than derive, as a mathematical proof, that for an arbitrary linear transformations there will be always an cartesian coordinate that $D-2$ axis are preserved under that transformation, where $D$ is the dimension. $\endgroup$ – Nogueira Jan 16 '18 at 19:40

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