Modifying select off-diagonal items in a matrix

This is something I have had the occasion to do, and never remember how, so this is legitimately a reminder to my future self of how to do things with off-diagonal elements of a matrix.

Select rows and columns are easy: mat[1:10, ] or mat[, -c(5)], for example, as is the diagonal: diag(mat) <- 0.

However, access to select off-diagonal slices is not obvious, without a loop. Fortunately, some friendly experts at Stack Overflow provide an excellent solution, which I include below with an invented application.