If $\pi_1:E_1\to M$ and $\pi_2:E_2\to M$ are two vector bundles over $M$, then we can construct a new vector bundle $E_1\oplus E_2$ by declaring the fibre at each $x\in M$ to be $(E_1\oplus E_2)_x:=(E_1)_x\oplus(E_2)_x$. One usually calls this (to my knowledge) the Whitney sum of $E_1$ and $E_2$. One could denote this by $E_1\times E_2$, but this seems rather unnecessary.
My question: if I encounter the notation $E_1\times E_2$, could any other vector bundle than the Whitney sum bundle be meant? The question popped up when considering Dirac structures on a manifold $M$, and the considering two Dirac structures on $M$ and their product $L_1\times L_2$.