  An apparent paradox in Einstein's Special Theory of Relativity, known as a
Thomas precession rotation in atomic physics, has been verified experimentally
in a number of ways. However, somewhat surprisingly, it has not yet been
demonstrated algebraically in a straightforward manner using
Lorentz-matrix-algebra. Authors in the past have resorted instead to computer
verifications, or to overly-complicated derivations, leaving undergraduate
students in particular with the impression that this is a mysterious and
mathematically inaccessible phenomenon. This is surprising because, as shown in
the present note, it is possible to use a basic property of orthogonal Lorentz
matrices and a judicious choice for the configuration of the relevant inertial
frames to give a very transparent algebraic proof. It is pedagogically useful
for physics students particularly at undergraduate level to explore this. It
not only clarifies the nature of the paradox at an accessible mathematical
level and sheds additional light on some mathematical properties of Lorentz
matrices and relatively-moving frames. It also illustrates the satisfaction
that a clear mathematical understanding of a physics problem can bring,
compared to uninspired computations or tortured derivations.
