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Let $F$ be a field. A $2$-$(7, 3, 1)_F$-subspace design, or $q$-Fano plane,over $F$, is a $7$-dimensional vector space $V$ over $F$ together with acollection $\mathfrak{B}$ of three-dimensional subspaces of $V$ such that everytwo-dimensional subspace of $V$ is contained in exactly one element $B$ of$\mathfrak{B}$. The question of existence of $q$-Fano planes over any field hasbeen open since the 1970s and has attracted considerable attention in thespecial case that $F$ is finite. Here we show the existence of $2$-$(7, 3,1)_F$-subspace designs over certain infinite fields $F$, including (amongothers) $\mathbb{Q}, \mathbb{R}$ and $\mathbb{F}_q(x, y, z)$ for $q$ odd.  The space $V$ is the 7-dimensional space of imaginary elements in a Cayleydivision algebra $O$ over $F$ and $\mathfrak{B}$ consists of the intersectionswith $V$ of all 4-dimensional subalgebras of $O$. We will present the requiredbackground on Cayley algebras in a self-contained fashion.  Next we study what happens if we apply the same procedure to split (ratherthan division) Cayley algebras. By identifying all four-dimensional subalgebrasof these, we show that in that case our construction still yields an inclusionminimal $(7, 3, 2)$ $q$-covering design. That is: every two-dimensionalsubspace of $V$ is contained in at least one element of the resulting set$\mathfrak{B}$ of three-dimensional subspaces of $V$ and no proper subset of$\mathfrak{B}$ has this property. However none of these $q$-covering designsare $q$-Fano planes. In the case that $F$ is finite we compute the number ofelements of $\mathfrak{B}$.  We also give a purely combinatorial construction of our $q$-Fano planes and$q$-covering designs for an abstract 7-dimensional $F$-vector space $V$ byidentifying the collection $\mathfrak{B}$ as a subvariety of the Grassmanian$Gr_3(V)$ defined entirely in terms of the classical Fano plane.

Cayley algebras give rise to $q$-Fano planes over certain infinite fields and $q$-covering designs over others