A flawed theorem about regular languages I am struggling with this question for a very long time and just can't find the flaw. So I am given a false Theorem:

The language $\{awwa \mid w \in \{a,b\}^* \}$ is regular.

Well, that part is obvious, we can prove that the language is not regular using the pumping lemma.
The question I am asked is to find the flaw in the &quot;proof&quot; for this theorem:

Let
$\qquad L_1 = \{ aw \mid w \in \{a,b\}^* \}$
$\qquad L_1^R = \{ wa \mid w \in \{a,b\}^* \}$
Let $L_2 = L_1 L_1^R = \{ awwa \mid w \in \{a,b\}^*\}$ be even-length palindromes that begin and end with an $a$.
Since $L_1$ is regular, and the class of regular languages is closed under reversal and concatenation, we conclude $L_2$ is also regular.

Can you find the flaw? I could build a DFA for $L_1$ and $L_1^R$, so I know they are regular. And regular languages are closed under reversal and concatenation.
But, however, $L_2$ is still not regular, so where is the mistake in the &quot;proof&quot;?
The concatenation of $L_1$ and $L_1^{\mathrm{R}}$ is not $\{awwa\mid w\in\{a,b\}^*\}$.