I want find all normal lie subgroups of $Iso(2)$. So i starting from finding normal subgroups of connected component of unity: $Iso^{\circ}(2)$, which corresponds to ideals of $\mathfrak{iso}(2)$. My attempt:
$\mathfrak{iso}(2) = \langle A,B,C| [A,B] = 0, [A,C]=-B, [B,C]=A\rangle$
So let $I = \langle \alpha_1 A + \beta_1 B = \gamma_1 C,\alpha_2 A + \beta_2 B = \gamma_2 C\rangle$ be ideal so $[A,I] = \langle -\gamma_1 B, -\gamma_2 B\rangle, [B,I] = \langle \gamma_1 A,\gamma_2 A\rangle, [C,I] = \langle \alpha_1 B - \beta_1 A, \alpha_2 B - \beta_2 A\rangle$. So if $\gamma_1 \neq 0$ or $\gamma_2 \neq 0$ then $A,B \in I \implies C \in I$. If $\gamma_1 \gamma_2 = 0$ then it's ideal and corresponded subgroups is subgroups in translations. Am i right? And if yes then how can i determine all normal subgroups?
I'm sorry for such stupid question, but i'm new in lie groups.
Note that any normal subgroup $H$ of $Iso(\mathbb{R}^2)$ contains all translations (here $H$ is not assumed to be closed).
For the second part, you are right - the translations form an abelian normal subgroup $T(2)$ of finite index in $Iso(\mathbb{R}^2)$. They correspond to the $2$-dimensional abelian ideal $I=\langle A,B\rangle$ in the Lie algebra $\mathfrak{iso}(\mathbb{R}^2)$.