Keywords: 14F05; 14E08; 11E08; 11E20; 11E88; 14F22; 14J26; 14M17; 15A66; Quadric; Intersection of quadrics; Derived category; Semiorthogonal decomposition; Clifford algebra; Morita theory; Brauer group; Rationality; Del Pezzo surface; Fano threefold;