The first column (or row, since it is symmetric) of a correlation matrix is unrestricted under the LKJ transformation (as distinct from the LKJ probability distribution). Thus, so is its Cholesky factor. Here is an example with the Cholesky factor of a 3x3 correlation matrix:

L = \begin{bmatrix}
1 & 0 & 0 \\
a & \sqrt{1 - a^2} & 0\\
b & c & \sqrt{1 - b^2 - c^2}
\end{bmatrix}

So, if \boldsymbol{\Sigma} = \mathbf{L}\mathbf{L}^\top, then \Sigma_{ij} is the dot product of the i-th and j-th rows of \mathbf{L}. If you want to restrict either \Sigma_{21} or \Sigma_{31} to be zero, that is easy: Just impose a = 0 or b = 0 respectively. However, technically that restriction means it is not LKJ prior any more, so you should not do `L ~ lkj_cor_cholesky(eta);`

. If you wanted to impose the restriction that \Sigma_{32} = ba + c\sqrt{1 - a^2} + 0\sqrt{1 - b^2 - c^2} = 0, that is a bit more complicated. If you already have a and b, then c = -\frac{ba}{\sqrt{1 - a^2}}, but then L_{33} = \sqrt{1 - b^2 - \frac{b^2a^2}{1 - a^2}} might not be real, depending on what b and a are.