A timelike (T L) constant axis ruled surface in E3
1 (Minkowski 3-space), as determined
by its ruling, forms a constant dual angle with its Disteli-axis (striction axis or curvature axis). In
this article, we employ the symmetry through point geometry of Lorentzian dual curves and the line
geometry of T L ruled surfaces. This produces the capability to expound a set of curvature functions
that specify the local configurations of T L ruled surfaces. Then, we gain some new constant axis
ruled surfaces in Lorentzian line space and their geometrical illustrations. Further, we also earn
several organizations among a T L constant axis ruled surface and its striction curve.