The collection of marked unimodular lattices in the euclidean $n$-space corresponds to the symmetric space $S_n:=SO_n(\mathbb{R}) \backslash SL_n(\mathbb{R})$.

I have only recently been made aware that $S_n$ is a so-called $linear$ symmetric space. Apparently this means the following: we can identify $S_n$ as a convex cone in some copy of $\mathbb{R}^{N+1}$ which contains no line through the origin and has nonempty interior. This identification is the correspondence between (marked) unimodular lattices and positive definite unimodular quadratic forms. Then the ''linearity'' of $S_n$ consists in that we have a group homomorphism $i: SL_n(\mathbb{R}) \to GL_{N+1}(\mathbb{R})=Aut \mathbb{R}^{N+1}$ for which the above identification is equivariant. Explicitly this homomorphism takes a matrix $A$ in $SL_n$ and $i(A)$ is acts on a positive definite quadratic form by $q$ by $i(A).q: x \mapsto q(Ax)$.

But I do not know what the ''linearity'' of $S_n$ $really$ means.

I would like to know how it relates to the following very remarkable property of unimodular lattices:

Take an unmarked rank $n$ unimodular lattice $\Gamma$. (ie. $\Gamma$ is an element in $S_n / SL_n(\mathbb{Z})$). The remarkable fact is this: for a given rational subspace $W$ in $\Gamma \otimes \mathbb{R}$ there is a CANONICAL complementary rational subspace $W^o$ --( of course, $W^o$ is just the orthogonal complement of $W$ in $\Gamma \otimes \mathbb{R}$).

My question is then how do we see the LINEARITY of $S_n$ as responsible for the fact that rational subspaces have CANONICAL rational complements? Any remarks on what a ''linear'' symmetric space really $is$ would be appreciated.