= Solution
The general <thin gravitational lens equation> is a two-component angular relation,
$$
\boldsymbol\beta=\boldsymbol\theta-\boldsymbol\alpha(\boldsymbol\theta),\qquad
\boldsymbol\alpha=\frac{D_{LS}}{D_S}\boldsymbol\xi.
$$
For a spherical lens, image, source and lens centre lie in one plane containing the line of sight. Choose its sky direction as a signed axis. Then $\alpha,\theta,\beta$ can be signed scalars and
$$
\boxed{\beta=\theta-\frac{D_{LS}}{D_S}\,\xi\,\operatorname{sgn}\theta.}
$$
They are not fundamentally scalar quantities for a general lens; the reduction uses this symmetry and axis choice. An exactly aligned source requires the full ring geometry, not a unique preferred sky axis.
In general, $D_S\ne D_L+D_{LS}$. For example, in a spatially flat expanding cosmology, comoving radial distances $\chi_L,\chi_S$ give
$$
D_L=\frac{\chi_L}{1+z_L},\qquad
D_S=\frac{\chi_S}{1+z_S},\qquad
D_{LS}=\frac{\chi_S-\chi_L}{1+z_S}.
$$
Thus $D_S-D_{LS}=\chi_L/(1+z_S)$ differs from $D_L$ when the redshifts differ. Ordinary Euclidean addition applies only when the same static physical-distance convention is appropriate.
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