Jacobi coordinates for three particles

For three particles, choose the relative position of particles one and two, the position of their center of mass relative to particle three, and the total center of mass. The kinetic energy diagonalizes with masses

For Jacobi coordinates $\mathbf{a},\mathbf{b},\mathbf{c}$ and their corresponding reduced masses $\alpha ,\beta ,\gamma$, total angular momentum is