In Special Relativity, if I have two linearly independent lightlike vectors is there a canonical way to construct a pair of orthogonal timelike and spacelike vectors?
For example, from u and v of lightcone coordinates we can make t and x easily...But that's simply because u and v are of that form...Is there a method for any arbitrary lightlike vectors?

\begin{eqnarray}
u_0 &=& v_0\\
u_1 &=& v_1 - \frac{\langle u_0, v_1 \rangle}{\langle u_0, u_0 \rangle} u_0\\
u_2 &=& v_2 - \frac{\langle u_0, v_2 \rangle}{\langle u_0, u_0 \rangle} u_0 - \frac{\langle u_1, v_2 \rangle}{\langle u_1, u_1 \rangle} u_1\\
u_3 &=& v_3 - \frac{\langle u_0, v_3 \rangle}{\langle u_0, u_0 \rangle} u_0 - \frac{\langle u_1, v_3 \rangle}{\langle u_1, u_1 \rangle} u_1 - \frac{\langle u_2, v_3 \rangle}{\langle u_2, u_2 \rangle} u_2\\
\end{eqnarray}

and then normalize it with \begin{equation}
e_i = \frac{u_i}{\| u_i \|}
\end{equation}

And now do the same calculation for the force on the opposite side of the loop. Now we get:
F = μ₀I/2𝜋(d+a) i d𝓁
and the force is in the opposite direction. Yes?

So if we combine these two elements there is a net force:
Fnet = μ₀I/2𝜋(d-a) i d𝓁 - μ₀I/2𝜋(d+a) i d𝓁
towards the wire.