The 3DPGA cheat sheet states the norm of a euclidean line to be \lVert \ell \rVert = \sqrt{\ell^2} = \sqrt{d^2+e^2+f^2}.

When I use \ell = a{\bf e_{01}} + b{\bf e_{02}} + c{\bf e_{03}} + d{\bf e_{12}} + e{\bf e_{31}} + f{\bf e_{23}} and compute \ell^2 I get \ell^2 = -d^2-e^2-f^2 + 2(af + be + cd){\bf e_{0123}} which as the Section 8.1.3 of the SIGGRAPH lecture note mentions is a dual number. Even though I haven’t read it anywhere (?) I assume that all lines are simple bivectors (since they can be seen as the intersection of two planes and the product of two planes always results in a simple bivector), and similarly the Plücker quadric relation equals the pseudoscalar part being zero. Nevertheless, the norm then should be \sqrt{-\ell^2} or \sqrt{|\ell^2|}, right?

Bonus question: If not all bivectors are lines, what do non-simple bivectors represent then in 3D PGA?