Find vectors $x\in\R_1^4$ in the previous examples such that $F=x^\perp$.
We have so solve the homogeneous system of equations: $\la b_j,x\ra=0$ for $j=1,2,3$. We only consider the first example. Put $x=\sum x_je_j$, then $\la b_1,x\ra=-x_0+2x_3=0$, $\la b_2,x\ra=x_1+2x_2=0$ and $\la b_3,x\ra=-x_0-2x_1-x_3=0$. Employing sage this can be solved as follows:
A=matrix([[-1,0,0,2],[0,1,2,0],[-1,-2,0,-1]])
A.right_kernel()
which will give you the answer $[8,-6 ,3 ,4]$, i.e. $x=8e_0-6e_1+3e_2+4e_3$ and this vector is time-like!