Does orthogonality imply linear independence for null vectors in the Minkowski space?

Question

In the Minkowski space with the metric $\text{diag}(-1,1,1,1)$, there exist vectors whose lenght is zero, but itself is nonzero. In this space, for four null and orthogonal vectors $a$, $b$, $c$, $d$, does the orthogonality imply their linear independence? I tried the usual method with the usual inner product of the Euclidean space but the condition that they are null vectors is a snag...

Answer 1

No. For example you can take $a=b=c=d$ for some vector $a\neq 0$ but with length zero, like $a=(1,1,0,0)$.