Generators of SO(8) group have been described by using direct product of the Gamma matrices and the Pauli Sigma matrices. We have obtained these generators in terms of generalized split octonion also. These generators have been used to describe the r
otational transformation of vectors for SO(7) symmetry group.
The 8 $times$ 8 matrix representation of SO(8) Symmetry has been defined by using the direct product of Pauli matrices and Gamma matrices. These 8 $times$ 8 matrices are being used to describe the rotations in SO(8) symmetry. The comparison of 8$time
s$8 matrices with octonions has also been shown. The transformations of SO(8) symmetry are represented with the help of Octonions and split Octonions spinors.
We have studied a sample of 415 associated (z_ab z_em; relative velocity with respect to QSO <3000km/s) Mg II absorption systems with 1.0<=z_ab<=1.86, in the spectra of SDSS DR3 QSOs, to determine the dust content and ionization state in the absorber
s. We studied the dependence of these properties on the properties of the QSOs and also, compared the properties with those of a similarly selected sample of 809 intervening systems (apparent relative velocity with respect to the QSO of >3000km/s), so as to understand their origin. From the analysis of the composite spectra, as well as from the comparison of measured equivalent widths in individual spectra, we conclude that the associated Mg II absorbers have higher apparent ionization, measured by the strength of the C IV absorption lines compared to the Mg II absorption lines, than the intervening absorbers. The ionization so measured appears to be related to apparent ejection velocity, being lower as the apparent ejection velocity is more and more positive. There is clear evidence, from the composite spectra, for SMC like dust attenuation in these systems; the 2175AA absorption feature is not present. The extinction is almost twice that observed in the similarly selected sample of intervening systems. We reconfirm that QSOs with non-zero FIRST radio flux are intrinsically redder than the QSOs with no detection in the FIRST survey. The incidence of associated Mg II systems in QSOs with non-zero FIRST radio flux is 1.7 times that in the QSOs with no detection in the FIRST survey. The associated absorbers in radio-detected QSOs which comprise about 12% of our sample, cause 3 times more reddening than the associated absorbers in radio-undetected QSOs. This excess reddening possibly suggests an intrinsic nature for the associated absorbers in radio-detected QSOs.
