Why is encrypting a key under itself dangerous?
Let $E$ be your cipher.
Consider the following cipher:
$$E_{k}'(m) = \begin{cases}
k &amp; m =k\\
E_k(m) &amp; \text{else}
\end{cases}$$
I believe you should be able to reduce the security of $E'$ to the security of $E$ in a rather straightforward manner.

This is a fairly important notion in general though.
The notion of "being secure, even if one gets encryptions of functions of the secret key" is known as Key Dependent Messaging security in general.
It's a property that (among other things) so far we seem to require to build FHE, but cannot prove that particular FHE schemes have it (instead we generally make the "Circular Security assumption").

Matthew Green has a blog post on it if you want a slightly higher level viewpoint on the topic.