Statement I: Total kinetic energy of a system equals i∑21mivi2. This is TRUE by definition.
Statement II: Total KE can be decomposed as KEtotal=21Mvcm2+i∑21mivi′2, where the first term is kinetic energy of center of mass motion and the second is kinetic energy of particles relative to the center of mass. This is TRUE by the energy decomposition theorem (Koenig's theorem).
Both statements are true.