<p>In a Banach algebra <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal{A}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>, for a pair of elements, the spectrum always commutes:<Equation ID="Equ1"> <EquationSource Format="TEX">\(\sigma(ab)\setminus\{0\}=\sigma(ba)\setminus\{0\},\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> <mo>=</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mi>a</mi> <mo stretchy="false">)</mo> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> <mo>,</mo> </mrow> </math></EquationSource> </Equation> whereas the exponential spectrum does not, as shown by Klaja and Ransford. In literature, the operator algebra <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal{B}(\ell^p\oplus\ell^q)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <msup> <mi>ℓ</mi> <mi>p</mi> </msup> <mo>⊕</mo> <msup> <mi>ℓ</mi> <mi>q</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> was anticipated to be a potential algebra of the same type. But we have established that the exponential spectrum commutes in this algebra. This is achieved by proving that the exponential spectrum commutes in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathcal{B(X)}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> if it commutes in the corresponding Calkin algebra.</p>