Normal series and refinements

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I am trying to show that no nontrivial refinements exists in the following series but haven't been able to make any progress: $\{1\}\unlhd Z(SL_2(\mathbb{R}))\unlhd SL_2(\mathbb{R})\unlhd Z(GL_2(\mathbb{R}))\unlhd GL_2(\mathbb{R})$ Is there a common technique of dealing with such kind of questions?