Generate matrices that satisfy multiple conditions

General Usage
2

This is very similar in spirit to your other question, trying to numerically “verify” something that you can show using math.

This is a very unusual application, and it is not possible to help you more without some context.

Specifically, is this a homework problem?

As you noted, if you randomly generate an orthonormal matrix, it has a 0 (theoretically, maybe near-0 numerically) chance of being triangular. Your approach is conceptually similar to

while true
    r = randn()
    if iszero(r)
        @info r^2 == r # verify that 0*0 == 0 numerically
    end
end