How to find the magnitude of electric and magnetic fields in an arbitary inertial frame?

In the context of special relativity, and using proper Lorentz transformations:

If the electromagnetic fields have an uniform value [= constant in space and time] in an inertial frame, how will the magnitude of these fields change in a different (arbitrary) inertial frame?

For example, an inertial frame which makes an angle $\alpha$, or moves with a different velocity $v'$ with respect to the original one.

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The relevant transformations, where ${\bf v}$ is a velocity (vector) of a "primed" frame with respect to the stationary "lab frame" and $\gamma$ is the Lorentz factor at speed $v$, are
$$\mathbf {{E}}' = \mathbf{{E}_{\parallel}} + \gamma \left( \mathbf {E}_{\bot} + \mathbf{ v} \times \mathbf {B} \right)$$ $$\mathbf {{B}}' = \mathbf{{B}_{\parallel}} + \gamma \left( \mathbf {B}_{\bot} -\frac{1}{c^2} \mathbf{ v} \times \mathbf {E} \right),$$ where the parallel and perpendicular subscripts refer to components of the E- and B-field that are either parallel or perpendicular to ${\bf v}$. i.e. The components parallel to ${\bf v}$ end up unchanged but the perpendicular components are transformed.