8 the temperature correlations in presence of a uniform magnetic field as discussed in [29] (see equation A3) are given by the CMB anisotropy in the presence of a homogeneous magnetic field at last scattering. The SH coefficients a$_{ℓm}$ can be expressed in terms of the CMB brightness fluctuations at last scattering using equation (13) and the generalized evolution equation (31) as 〈 Δ( ⃗ k, ˆ n, τ$_{s}$ )Δ( ⃗ k, ˆ n $^{′}$, τ$_{s}$ ) 〉 ∝ ˜ Δ 1 M 11 ( k, τ$_{s}$ ) ˜ Δ 1 M ′ 11 ( k, τ$_{s}$ ) × (ˆ n × ˆ k ) · ( ˆ n ′ × ˆ k ) . (46) a$_{ℓm}$ = 4 π ∫ d $^{3}$k (2 π ) 3 φ ( ⃗ k ) ∑ ℓ$_{1}$m$_{1}$LM β$_{ℓ}$$_{1}$$_{ℓ}$ C LM ℓ$_{1}$m$_{1}$ℓm Y$_{ℓ}$$_{1}$$_{m}$$_{1}$ ( ˆ k ) × ∑ ℓ$_{2}$ℓ$_{3}$ℓ$_{4}$ C ( L, ℓ$_{2}$, ℓ$_{1}$, ℓ ) j$_{ℓ}$$_{2}$ ( k Δ τ ) ˜ Δ LM ℓ$_{3}$$_{ℓ$_{4}$}$( k, τ$_{s}$ ) , (41) a$_{ℓm}$ = 4 π ∫ d $^{3}$k (2 π ) 3 φ ( ⃗ k ) ∑ m$_{1}$M β$_{1}$$_{ℓ}$ C 1 M m$_{1}$ℓm Y$_{1}$$_{m}$$_{1}$ ( ˆ k ) × ∑ ℓ$_{3}$ [( ... ) j$_{ℓ}$$_{−}$$_{1}$ ( k Δ τ ) + ( ... ) j$_{ℓ}$$_{+1}$ ( k Δ τ )] × ˜ Δ 1 M 11 ( k, τ$_{s}$ ) . (47) Here the bipolar dipole terms of the CMB brightness fluc- tuation ˜ Δ 1 M 11 ( k, τ$_{s}$ ) ˜ Δ 1 M ′ 11 ( k, τ$_{s}$ ) in equation (44), encap- sulates the source term due to the presence of a uniform magnetic field [29]. In equation (41), the bipolar dipole terms ˜ Δ 1 M 11 ( k, τ$_{s}$ ) gives rise to SH coefficients where C ( L, ℓ$_{2}$, ℓ$_{1}$, ℓ ) is defined in equation (32). Hence, in general 〈 a$_{ℓm}$a$_{ℓ}$ ′ m ′ 〉 correlations measured at present are related to 〈 ˜ Δ LM ℓ$_{1}$$_{ℓ$_{2}$}$( k, τ$_{s}$ ) ˜ Δ L $^{′}$M ′ ℓ$_{3}$ℓ$_{4}$ ( k, τ$_{s}$ ) 〉 correla- tion between the generalized Boltzmann fluctuations at the last scattering surface. In particular, SI violation encoded in the off-diagonal cor- relation 〈 a$_{ℓm}$a$_{ℓ}$ ′ m ′ 〉 (and non-zero BipoSH A LM ℓℓ ′ , L > 0) are related as The angular correlations in this case turn out to be as in equations (24, 25, 26) or more generally for different Bipolar coefficients J ′ and N ′ as in equation (22) when the power spectrum is also anisotropic. Using equation (11), correlations of the CMB brightness fluctuations are It is interesting to note that the known diagonal ( ℓ ′ = ℓ ) and off-diagonal ( ℓ ′ = ℓ ± 2) correlations in presence of a homogeneous magnetic field [29, 30], can be easily recovered in our approach. Work is in progress to relate other cases of SI violation originating in the physics at the last scattering surface using this formalism. 〈 a$_{ℓm}$a ∗ ℓ ′ m ′ 〉 = ( ... ) δ$_{ℓ}$ ′ $_{ℓ}$δ$_{mm}$ ′ + ( ... ) δ$_{ℓ}$ ′ ℓ ± $_{2}$δ$_{mm}$ ′ . (48) A JN ℓℓ ′ ∼ { ˜ Δ L ℓ$_{1}$$_{ℓ}$( k, τ$_{s}$ ) ⊗ ˜ Δ L ′ ℓ$_{2}$ℓ ′ ( k, τ$_{s}$ ) } JN (42) 〈 Δ( ⃗ k, ˆ n, τ$_{s}$ )Δ( ⃗ k, ˆ n $^{′}$, τ$_{s}$ ) 〉 = (4 π ) 2 ∑ ℓ$_{1}$ℓ$_{2}$LM ∑ ℓ$_{3}$ℓ$_{4}$L ′ M ′ β$_{ℓ}$$_{1}$$_{ℓ}$$_{2}$ β$_{ℓ}$$_{3}$$_{ℓ}$$_{4}$ × ˜ Δ LM ℓ$_{1}$$_{ℓ$_{2}$}$( k, τ$_{s}$ ) ˜ Δ L $^{′}$M ′ ℓ$_{3}$ℓ$_{4}$ ( k, τ$_{s}$ ) × { Y$_{ℓ}$$_{1}$ ( ˆ k ) ⊗ Y$_{ℓ}$$_{2}$ (ˆ n ) } LM × Y$_{ℓ}$$_{3}$ ( ˆ k ) ⊗ Y$_{ℓ}$$_{4}$ ( ˆ n $^{′}$) L ′ M ′ . (43) The search for subtle statistical isotropy breakdown in the universe is highly motivated by numerous theoreti- cal scenarios. The fluctuations in the cosmic microwave background is the arguably the most promising observa- tional probe of the SI of the universe. The violation of SI could have its origin not only in in anisotropic primor- dial power spectrum, but also in the SI violation in the fluctuations of the baryon-photon fluid at last scattering. SI deviations generated by a general form of anisotropic primordial power spectrum for isotropic Boltzmann func- tions has been studied in the recent literature [42]. This paper includes this equally important possibility of a gen- eral SI breakdown in the CMB photon distribution func- tion. We study the generalized case of SI violation in terms of Bipolar spherical harmonic (BipoSH) bright- ness fluctuations, substantially extending the scope of origin of SI violation solely from the anisotropic primor- dial power spectrum. We illustrate the generality and power of our formalism using the case for a uniform magnetic field. We show the correlations of the CMB brightness fluctuations in this particular case is sourced by the bipolar dipole ( L = 1) terms of equation (11) with ℓ$_{1}$ = ℓ$_{2}$ = 1 where V. CONCLUSIONS Δ( ⃗ k, ˆ n, τ$_{s}$ ) = 4 $^{$_{πi}$√$_{3}$}$∑ M ˜ Δ 1 M 11 ( k, τ$_{s}$ ) { Y$_{1}$ ( ˆ k ) ⊗ Y$_{1}$ (ˆ n ) } 1 M = 3 i √ 3 2 ∑ M ˜ Δ 1 M 11 ( k, τ$_{s}$ ) ( ˆ k × ˆ n )$_{M}$ , (44) (ˆ n · ˆ n $^{′}$)( ˆ k · ˆ k ) − (ˆ n · ˆ k )( ˆ n ′ · ˆ k ) = (ˆ n × ˆ k ) · ( ˆ n ′ × ˆ k ) , (45) The breakdown of SI in the CMB brightness fluctu- ation results in off-diagonal terms in the SH space an- gular correlations 〈 a$_{ℓm}$a$_{ℓ}$ ′ m ′ 〉 , or, equivalently, in the co- efficients the Bipolar spherical harmonic (BipoSH) rep- resentation [38, 39]. We relate the measurable BipoSH with M = {− 1 , 0 , +1 } . Here ( ˆ k × ˆ n )$_{M}$ is the usual cross- product written as irreducible products of the rotation group [48]. Using standard vector identity [48] and equation (8)