I think I understand why the string $( [ ) ]$ is not in a Dyck language.
In my words,
D2* is all the dyck words of 2 parentheses.
From the definiton of $D2*$, every words must have exactly 2 pairs correctly nested pairs of parentheses.
All the words of the language are given by
$\sum = { (1, )1, (2, )2 } $
The alphabet consisting of 2 kinds of left and right parentheses.
The language is given by the following grammar :
Terminal set is $\sum$
Start variable is not in terminal set
$ S -> \big( \epsilon | SS | (i S )i \big)$ for all i > 1
But the form of that word is $(i (j )i )j$ which fails to match the definition.
But I'm having trouble writing a formal proof on that.