From an older exercise sheet in my relativity class which I completed, but didn't annotate enough to remember it everything was correct:
Let $X$ be an inertial observer noticing that a spacetime event $B$ was caused by another event $A$. Show that no inertial observer $X'$ exists for whom the event $A$ is seen to be caused by $B$.
I chose the following ansatz:
Let $t_B>t_A$ in $X$. Then the Lorentz transformation of $t_A$ and $t_B$ gives us $$ t_A' = \frac{ct_A - \beta x_A}{\sqrt{1-\beta^2}} \quad t_B' = \frac{ct_B - \beta x_B}{\sqrt{1-\beta^2}} $$ Now, for causality to be given, we demand $0 < t_B - t_A$, i.e., the event $B$ happens after $A$. $$0 < t_B - t_A = \gamma (ct_B - \beta x_B - ct_A + \beta x_A)\\ =\gamma [c(t_B - t_A) - \beta (x_B - x_A)]$$ Losing the gamma and rearranging gives us $$ \beta (x_B - x_A) < c(t_B - t_A) \iff v/c^2 (x_B - x_A) < t_B - t_A $$ Now, since $v<c$, the expression $t_B - t_A \geq \frac{x_B - x_A}{c}$ holds.
I haven't written anything else after this, and I'm not quite sure whether a) this proves the causality-independence of the observer or b) my way of calculating is correct.
Looking forward to answers.