A characterisation of Baer subplanes

Let $${{\mathcal {K}}}$$ be a set of $$q^2+2q+1$$ points in $$\text {PG}(4,q)$$. We show that if every 3-space meets $${{\mathcal {K}}}$$ in either one, two or three lines, a line and a non-degenerate conic, or a twisted cubic, then $${{\mathcal {K}}}$$ is a ruled cubic surface. Moreover, $${{\mathcal {K}}}$$ corresponds via the Bruck–Bose representation to a tangent Baer subplane of $$\text {PG}(2,q^2)$$. We use this to prove a characterisation in $$\text {PG}(2,q^2)$$ of a set of points $${{\mathcal {B}}}$$ as a tangent Baer subplane by looking at the intersections of $${{\mathcal {B}}}$$ with Baer-pencils.