Suppose $b_1,\ldots,b_n$ is a basis for $E$ such that none of the spaces $E_k\colon=\lhull{b_1,\ldots,b_k}$, $k=1,\ldots,n$, is degenerated. Then the Gram-Schmidt algorithm terminates successfully.
Since $E_k$ is not degenerated the orthpogonal projektion $P_k:E\rar E_k$ is defined. The vector $x\colon=b_{k+1}-P_kb_{k+1}$ is orthogonal to $E_k$ and not equal to $0$, thus $\lhull{E_k,x}=E_{k+1}$. If $x$ is light-like, then $x\perp x$ and therefore $x\perp E_{k+1}$, i.e. $E_{k+1}$ is degenerated.