A similar question was already asked in question titled "Spectra of sums and products in (Banach) algebras [was: Spectrum in Banach Algebra]".

Answer there led me to the following question.

If for elements $a,b$ in Banach algebra $A$ hold that $\operatorname{spec}(\lambda a+\mu b)\subseteq \lambda \operatorname{spec}(a)+\mu \operatorname{spec}(b)$ for every $\lambda, \mu\in \mathbb{C}$, can we say something about the subalgebra $B$ of $A$ generated by $a$ and $b$. Might we conclude something about $B/\operatorname{rad}(B)$?

Where to look for (possible) counterexample of $B/\operatorname{rad}(B)$ being commutative?

Thank you for finding a counterexample. Now I would like to add some additional assumptions. Suppose that for a selfadjoint element $a$ and a unitary element $u$ of $C^∗$-algebra holds that $\operatorname{spec}(a+\lambda a^2)=\operatorname{spec}(a+\lambda ua^2u^∗)$ for every $λ∈\mathbb{C}$. Moreover, if $\lambda \in \mathbb{R}$ elements $a+\lambda a^2$ and $a+\lambda ua^2u^∗$ are unitarily equivalent.