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Suppose the axes in two coordinate systems S, S' are parallel. Now, suppose I rotate S through some angle $\theta$, and also rotate S' through the same angle $\theta$. It's not clear to me that the rotated axes will remain parallel. In relativity, angles are coordinate dependent. For example, if S and S' are parallel and S' moves at velocity |v|$\hat x$ relative to S, and someone in S' places a bar at an angle of $\theta$ with respect to the axis x', then this angle will not be the same as the angle relative to the x axis. If someone in S then places a bar at an angle $\theta$ with respect to x, then am I correct that this bar won't be parallel to the bar in S' since an observer in S measures the bars at different angles relative to the x axis? However, these bars are just rotated axes, both rotated at the same angle $\theta$, but they're not parallel.

user7348
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The Lorentz group contains both ordinary rotations and boosts. Any Lorentz transformation may be written generally in terms of an ordinary rotation and a pure boost: $\Lambda=RL=\tilde{R}\tilde{L}$ So that a Lorentz transformation From some third frame $K$, also parallel to $S$ and $S^\prime$, will be given as: $$\Lambda=RL=\tilde{L}\tilde{R};\;\;\text{For K to S}$$ and: $$\Lambda=R L^\prime=\tilde{R}\tilde{L^\prime};\;\; \text{For K to S}^\prime.$$ Because the $S$ and $S^\prime$ are rotated by the same amount, the same rotation $R$ is used in either case, thus it is obvious that if $S$ and $S^\prime$ were parallel in $K$ then the transformed frames are also parallel.

Albertus Magnus
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