Thrall states ("On the Projective Structure of a Modular Lattice", 1951, https://scholar.google.com/scholar?cluster=4998496641867146321):

One measure of the complexity of a modular lattice is how much it lacks being distributive. Each triple of elements in $L$ for which the distributive law does not hold generates a lattice which has a five element sublattice $P = \{r, s, t, u, w\}$ in which $r$ is the meet and $w$ is the join of each pair selected from $s, t, u$. We shall call such a five element lattice a

projective root(p.r.) and shall use the notation $P = [r; s, t, u; w]$ to indicate that $P$ is a projective root. If any one (and therefore all) of the six quotients [intervals] $s/r, t/r, u/r, w/s, w/t, w/u$ is prime [has height 1] in $L$, we say that $P$ is aprime projective root(p.p.r.) n $L$.It is easily seen (we omit the proof) that if $L$ contains any projective root, then it contains a prime projective root. Indeed, if $P = [r;s,t,u;w]$ is a p.r. with dimension $s/r = k$, then $L$ contains $k$ p.p.r.'s $P_i = [r_i; s_i, t_i, u_i; w_i], i= 1, \ldots, k$, such that $r=r_1, w_1 = r_2, \ldots, w_{k-i} = r_k, w_k = w$.

My assumption is that the proof takes $P$ and dissects out of the interval $[r, w]$ such a chain of $P_i$. So the value of the result is showing that the signatures of non-distributivity, the p.r.s, are constructed from p.p.r.s.

I'm just learning lattice theory and I can't prove this result, or even see what the intended method of attack is. I've checked Google Scholar, and none of the references seem to pay attention to this result. Similarly, I've looked at a few of the standard references, and none of them discuss this result, or even this approach toward determining the "structure" of a non-distributive modular lattice.