Kronecker's delta is just another name for the unit matrix. In the unit matrix only the diagonal terms are unity and all off-diagonal terms are zero. Similarly for kronecker's delta delta_11=delta_22=delta_33=1 and delta_12=delta_21=delta_13=delta_31=delta_23=delta_32=0. Thus ai. a*j=delta_ij is just a short way of writing nine dot products a1.a*1=1; a1.a*2=0; a1.a*3=0; a2.a*1=0; a2.a*2=1; a2.a*3=0; a3.a*1=0; a3.a*2=1; a3.a*3=1; The above relations are true for all systems and not only for cubic system. Note that we are not requiring a1.a2=0 which is true for orthogonal system (cubic, tetragonal and orthorhombic) but not for a general parallelopiped. We are requiring a1.a*2=0 where a*2 is a basis vector of reciprocal lattice and is not the same as a2. So by the relation a1.a*2=0 we are requiring that the second basis vector (a*2) of the reciprocal basis is orthogonal to the first basis vector (a1) of the real lattice. Hope this clarifies.

This is purely a question of convention. The convention used by me is preferred by crystallographers where ai*aj=delta_ij. The convention often used by physicists is ai*aj*=2pi delta_ij. One can use either convention.

@@rajeshprasadlectures thank you a lot, now everything is clear

Crystals (lattice+motif) can have 32 point groups. Lattice alone can have only 7 point groups.

The conventional cell of the HCP structure is itself primitive.

The magnitude is the same but directions are different. Delta K is difference between two vectors.

Yes. ChatGPT is learning fast! In the end, I also provide it with the correct answer :)

Thank you Shahid