External direct product of formal group laws

Definition

Suppose $R$ is a commutative unital ring. Suppose $F_{1}$ and $F_{2}$ are formal group laws over $R$ of dimensions $d_{1}$ and $d_{2}$ respectively. The external direct product of $F_{1}$ and $F_{2}$, which we denote $F_{1}\times F_{2}$, is a formal group law of dimension $d_{1}+d_{2}$, given by:

$(F_{1}\times F_{2})(x_{1},x_{2},\dots ,x_{d_{1}},x_{d_{1}+1},\dots ,x_{d_{1}+d_{2}},y_{1},y_{2},\dots ,y_{d_{1}},y_{d_{1}+1},\dots ,y_{d_{1}+d_{2}})$

$=(F_{1}(x_{1},x_{2},\dots ,x_{d_{1}},y_{1},y_{2},\dots ,y_{d_{1}}),F_{2}(x_{d_{1}+1},\dots ,x_{d_{1}+d_{2}},y_{d_{1}+1},\dots ,y_{d_{1}+d_{2}}))$

In other words, the first $d_{1}$ coordinates of $F_{1}\times F_{2}$ are obtained by applying $F_{1}$ to the first $d_{1}$ coordinates of the two inputs, and the last $d_{2}$ coordinates of $F_{1}\times F_{2}$ are obtained by applying $F_{2}$ to the last $d_{2}$ coordinates of the two inputs.