Pasted

R.<t, u_,v_,w_, u,v,w, x,y,z> = QQ[]
 # aim: verify automatically that x^2 + y^2 - z^2 = -1 is composed of lines.
 p = x^2 + y^2 - z^2 + 1
 line = vector((x, y, z)) + t*vector((u, v, w))
 # for all x,y,z if p(x,y,z) = 0 then
# there are u,v,w not all zero such that
# for all t, p(line) = 0
 p_line = p(x=line[0], y=line[1], z=line[2])
line_included = forall_t_p_line = p_line.polynomial(t).coefficients()
 uvw_nonzero = u*u_ + v*v_ + w*w_ + 1
uvw_condition = ideal(line_included + [uvw_nonzero])
 there_are_uvw = uvw_condition.elimination_ideal([u,v,w,u_,v_,w_])
 assert there_are_uvw == ideal(p), "p does not appear to be composed of lines!"
 print('OK')