Background:
Associated to an abelian variety of dimension $n$ is a formal group law of dimension $n$, that is, a formal power series, $F(\bar{x}, \bar{y}) = \sum_{i, j} c_{ij} \bar{x}^i \bar{y}^j$, where $\bar{x} = (x_1, ...., x_n)$.
Further, this formal power series obeys $F(\bar{x}, F(\bar{y}, \bar{z})) = F(F(\bar{x}, \bar{y}), \bar{z})$ and $F(\bar{x}, 0) = \bar{x}$.
We may sometimes decompose a formal group law into a formal summand of two smaller formal group laws.
Here's my question: What does formal summand explicitly mean in this context?
Is it always a literal sum of power series? That is, $F((x_1, ..., x_n), (y_1, ..., y_n)) = (G((x_1, ..., x_k), (y_1, ..., y_k)), H((x_{k+1}, ..., x_n), (y_{k+1}, ..., y_n))$.
In the case where our abelian variety is a product of two abelian varieties $C = A \times B$, I assume the formal group law of $C$ will be the product of the formal group laws of $A$ and $B$. In this case, I believe that the formal decomposition is the coordinate decomposition listed above. In the case of the additive formal group law, $\mathbb{G}_a(A^2) = \mathbb{G}_a(A^1) \times \mathbb{G}_a(A^1)$, since $(x_1, x_2) + (y_1, y_2) = (x_1 + y_1, x_2 + y_2) $.
I am however desperately baffled by \textit{the case where we are decomposing the formal group law of an abelian variety via the method of complex multiplication}.
In this case, we decompose the characteristic $p$-reduction $p$-divisible group of the abelian variety and this decomposition imposes a splitting on the formal group law of the variety.
Here's my secondary question: Let $A$ be an abelian variety. How does the splitting of the $p$-divisible group of A impose a splitting of the formal group law of A?
The splitting of $p$-divisible group I am referring to is as follows. Let $\pi$ be the geometric Frobenius of $A$. Let $S$ be the set of discrete valuations of $\mathbb{Q}(\pi)$ dividing the prime number $p$. Then, $A[p^\infty] = \prod_{s \in S} A[p^\infty]_s$, where $A[p^\infty]_s$ denotes localization at $s$.
It seems highly unlikely that the splitting imposed on the formal group law of the variety is a simple coordinate splitting, and I cannot think of what else it could possibly be.
I have been informed that in the case of the p-divisible group splitting, there is an isomorphism (coordinate change) which is equivalent to the naive split. So, rather than $F = G \times H$, we have $F \simeq G \times H$.