Notation for the direct sum and Kronecker sum of matrices 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...

I've seen people use "$A+B$" for the Kronecker sum. Since $A$ and $B$ act on different spaces and so can't be added normally, the reader is supposed to think: "I need to view $A$ and $B$ as acting on the same space. Aha! So '$A$' must mean '$A\otimes I$' and '$B$' must mean '$I\otimes B$' so that '$A+B$' refers to '$A\otimes I+I\otimes B$'".
