The density of the LKJ cholesky is

which is from the Stan manual ( 24.2 Cholesky LKJ correlation distribution | Stan Functions Reference (mc-stan.org)). I get where the 2\eta - 2 comes from, it is

which implies that |J| = \prod_{k=2}^K L_{kk}^{K - k}. However, the Edelman handout ( see top of page 13 z_handout2.dvi (mit.edu))

has the jacobian for the Cholesky transform as

So where am I going wrong with this?