Two timelike vectors can be put in a two-dimensional timelike subspace

Theorem : For any two timelike vectors $u, v$, there exists a timelike two-dimensional subspace which contains them.

Proof : If $u$ and $v$ are collinear, simply define a one-dimensional timelike subspace and complement it with any orthogonal spacelike vector. Otherwise, consider some Gram-Schmidt basis built from $u$. Then, for $w$ a timelike vector,

\begin{equation} v = \alpha u + w \end{equation}

All of our vectors will therefore be contained in a subspace spanned by $u$ and $w$, which is a timelike subspace.

Corrolaries : For any two timelike vectors, there exists a basis in which $u^\mu = (u_t, u_x, 0, \ldots)$ and $v^\mu = (v_t, v_x, 0, \ldots)$. If we build our basis from $u$ or $v$, we get $u_x = 0$ or $v_x = 0$. We have two cases :

$u$ and $v$ have the same time-orientation