The tensor product $w = u \otimes v$ of two vectors $u \in \mathbb{R}^m$ and $v \in \mathbb{R}^n$ is usually defined as \begin{equation} w_{in + j} = u_i v_j \text, \end{equation} and the Kronecker product of two (appropriately-sized) matrices $A \otimes B$ acts on such elementary tensors as \begin{equation} (A \otimes B) (u \otimes v) = (A u) \otimes (B u) \text. \end{equation}
The direct sum $w' = u \oplus v$ of two vectors is usually defined as \begin{equation} w'_i = \begin{cases} u_i & \text{if $i \le m$,} \\ v_{i - m} & \text{if $m < i \le n$,} \end{cases} \end{equation} and the direct sum of two matrices acts on this sum as \begin{equation} (A \oplus B) (u \oplus v) = (A u) \oplus (B u) \text. \end{equation}
One can also define the Kronecker sum of square matrices \begin{equation} A \mathop{\oplus_{\mathrm{Kronecker}}} B = A \otimes I_n + I_m \otimes B \text, \end{equation} such that \begin{equation} (A \mathop{\oplus_{\mathrm{Kronecker}}} B) (u \otimes v) = (Au) \otimes v + u \otimes (Bv) \text. \end{equation} This last operation is interesting in e.g. the theory of continous-time Markov chains, where if $A$ and $B$ represent infinitesimal generator matrices of two non-interacting systems, then $A \mathop{\oplus_{\mathrm{Kronecker}}} B$ is the infinitesimal generator for the (asynchronous) composition of the two systems.
Unfortunately, both the direct sum and the Kronecker sum of matrices is denoted by $\oplus$. What are some "standard" ways to handle this notational confusion if one has to work with mixed direct sums and Kronecker sums of matrices? The notation \begin{equation} A \mathop{\oplus_{\mathrm{Kronecker}}} (B \oplus C) \end{equation} is very cumbersome.
Wikipedia claims (unfortunately, I didn't find the source) that sometimes $\boxtimes$ is used to denote Kronecker products. Analogously, I could use $\boxplus$ for Kronekcer sums. However, I would prefer to use $\otimes$ and $\oplus$ for the Kronecker operators, mostly for aesthetic reasons (the box $\square$ is very "pointy").
Another way would be to use alternative notation for the direct sum. I guess by a not-so-slight abuse of notation, I could use $\times$ for the direct sum of vectors. $u \times v \in U \times V$ does not look so bad, and I don't need to work with infinte products where the direct sum and product are actually different. On the other hand, this feels like a huge hack...