I have been learning about elliptic divisibility sequences, I started reading this article on Wikipedia (https://en.wikipedia.org/wiki/Elliptic_divisibility_sequence#:~:text=In%20mathematics%2C%20an%20elliptic%20divisibility,Morgan%20Ward%20in%20the%201940s).
Here, elliptic divisibility sequences are first defined by the recursions $$W_{2n+1}W_1^3 = W_{n+2}W_n^3 - W_{n+1}^3W_{n-1},\qquad n \ge 2, \tag{1}$$ $$W_{2n}W_2W_1^2 = W_{n+2}W_n W_{n-1}^2 - W_n W_{n-2}W_{n+1}^2,\qquad n\ge 3, \tag{2}$$ where $W_1, W_2$ and $W_3$ are non-zero integers.
In a later section, it is later claimed without any proof or reference that $$W_{n+m}W_{n-m}W_r^2 = W_{n+r}W_{n-r}W_m^2 - W_{m+r}W_{m-r}W_n^2 \quad\text{for all}\quad n > m > r. \tag{3}$$ I am quite curious about how (1) and (2) imply (3). With induction and some manipulations, I have been able to show (3) in general, but only by assuming it to be true for $r=1$, which can be written as $$W_{n+m}W_{n-m}W_1^2 = W_{n+1}W_{n-1}W_m^2 - W_{m+1}W_{m-1}W_n^2 \quad\text{for all}\quad n > m > 1. \tag{4}$$ Hence, I have been trying to show (4) for a very long time. I have tried various kinds of inductive reasoning: for instance, I could show it for $m=2$ (allowing $n>m$ to be arbitrary), and assuming it to be true up to $m-1$, I applied the inductive hypothesis on $(n+1, m), (n, m-1)$ and $(n-1, m)$ playing the role of $(n, m)$, from which I could get an expression for $W_{n+m} W_{n-m}$ in terms of $W_j$ with $j \in S_{n, m} := \{n-2, n-1, n, n+1, m-2, m-1, m, m+1\}$. However, this brings the induction step to showing a certain identity between the $W_j$ for $j \in S_{n, m}$, and it is not clear to me why such an identity should hold true. (It seems to have too few indices and hence perhaps too much freedom?)
All other references that I found either start by defining an elliptic divisibility sequence by (3) or by (the equivalent) (4), so I am starting to doubt whether (1) and (2) actually imply (4) at all. In that regard, as far as I know, it is true that the sequence of (appropriately-signed) square roots of the denominators of the abscissae (i.e., $x$-coordinates) of the positive-integral multiples of a point on an elliptic curve, do satisfy (1) and (2). However, I think I may have read somewhere (cannot recollect where exactly) that this sequence does not satisfy the general recursion (3), and hence cannot satisfy (4) either. Of course if this were the case, then (1) and (2) would not imply (4).
So my question is: do (1) and (2) imply (4) in general? If not, what minimal conditions are needed (on the initial terms and/or the indices $m, n, r$) for this implication to hold true? (It would be great if a reference could be found proving (3) in general, but I would certainly appreciate any input.)