Weyl, Space-Time-Matter: The Riemann "quadrilinear form", and "functions which stand in quadratic relationship with an element of surface" The following is from pages 83 and 84 of Space—Time—Matter by Hermann Weyl:

Sometimes conditions of symmetry more complicated than
those considered heretofore occur. In the realm of quadrilinear
forms $F(\xi, \eta, \xi', \eta')$ those play a particular part which satisfy the
conditions (Eqs 39)
$$\begin{aligned}
F(\eta, \xi,\xi',\eta')=& F(\xi,\eta,\eta',\xi') = -F(\xi,\eta,\xi',\eta'), \\
F(\xi',\eta',\xi,\eta)=& F(\xi,\eta,\xi',\eta'),\\
0=&F(\xi,\eta,\xi',\eta') + F(\xi,\xi',\eta',\eta) + F(\xi,\eta',\eta,\xi').
\end{aligned}$$
For it may be shown that for every quadratic form of an arbitrary
two-dimensional space-element
$$
\xi^{ik} = \xi^{i} \eta^{k} - \xi^{k} \eta^{i}
$$
there is one and only one quadrilinear form $F$ which satisfies
these conditions of symmetry, and from which the above quadratic
form is derived by identifying the second pair of variables $\xi'$, $\eta'$
with the first pair $\xi$, $\eta$. We must consequently use co-variant
tensors of the fourth order having the symmetrical properties (Eqs 39)
if we wish to represent functions which stand in quadratic relationship
with an element of surface.

It clearly foreshadows the Riemann curvature tensor, but I'm not sure how to read it.  Specifically I'm not sure what is meant by "quadratic form of an arbitrary two-dimensional space-element".  On pages 81 and 82 he discusses the squared magnitude of a two-dimensional space-element $\frac{1}{2}\xi^{ij}\xi_{ij}=\frac{1}{2}g_{ik}g_{jl}\xi^{ij}\xi^{kl}$.  That's my best guest as to what is intended.  I am similarly confused by "functions which stand in quadratic relationship with an element of surface."
If someone would shed some light on this, I would appreciate it.
 A: I think Weyl says the following: A quadratic form of an arbitrary two-dimensional space-element $\xi^{ik}=\xi^i\eta^k-\xi^k\eta^i$ is
$$
Q(\xi^{ik})=a_{jlmn}\xi^{jl}\xi^{mn}\quad\quad\text{ (with summation convention). }
$$
(Instead of writing $Q(\xi)$ I kept the indices in $\xi^{ik}$ to distiguish it from the one-dimensional $\xi^i\,.$)

*

*For every such $Q$ there is a quadrilinear form $F(\xi,\eta,\xi',\eta')$ such that $Q(\xi^{ik})=F(\xi,\eta,\xi,\eta)\,.$ This is trivial. Just define
$$\tag{1}
F(\xi,\eta,\xi',\eta')=a_{jlmn}\,\xi^{jl}(\xi')^{mn}\,.
$$


*The $F$ from (1) is unique. To see that let's write $F$ as a bilinear form
of two-dimensional space elements
$$
F(\xi,\eta,\xi',\eta')=Q(\xi^{ik},(\xi')^{ik})\,.
$$
From
$$\tag{2}
Q\big(\xi^{ik}+(\xi')^{ik},\xi^{ik}+(\xi')^{ik}\big)=
Q(\xi^{ik})+
2\,\underbrace{Q\big(\xi^{ik},(\xi')^{ik}\big)}_{F(\xi,\eta,\xi',\eta')}+
Q\big((\xi')^{ik})
$$
it follows that $F(\xi,\eta,\xi',\eta')$ is determined by the other three quadratic form terms in (2).
A more modern treatment of symmetry/anti-symmetry relationships of multilinear tensors is based on differential forms and their wedge product. The book Graviation of Misner, Thorne and Wheeler is my favourite reference.
