Proposal for correlation matrix with LKJ prior

I am writing a Gibbs sampler from scratch. As recommended in various places (http://www3.stat.sinica.edu.tw/statistica/oldpdf/A10n416.pdf, and in another question Covariance matrix proposal distribution) I have split covariance matrices for random effects (e.g. $$A$$) into a vector $$\sigma$$ of standard deviations and a correlation matrix $$\Omega$$ in the following way:

$$A=\text{diag_matrix}(\sigma)\text{ }\Omega \text{ diag_matrix}(\sigma)$$

I have assigned half student t priors to the standard deviations $$\sigma$$ and an LKJ prior to the correlation matrix $$\Omega$$ (see e.g. recommendations under Hyperpriors in https://mc-stan.org/docs/2_18/stan-users-guide/multivariate-hierarchical-priors-section.html). I wanted to update the correlation matrix $$\Omega$$ in a single block rather than update each unique element of it individually.

From my model, the posterior for updating the correlation matrix $$\Omega$$ does not have a standard form, and so I was considering using a Metropolis Hastings step to update it. However I'm not sure what proposal to use, because:

• Most examples I have seen suggest proposals for covariance matrices rather than correlation matrices
• The proposal will need to restrict the off-diagonals of the correlation matrix to fall between -1 and 1, and the on diagonals to equal 1
• I don't see how you could use an LKJ distribution as a proposal and make it rely on the current $$\Omega$$ when sampling the new $$\Omega'$$

What proposals would allow update of the correlation matrix in a single step, relying on the current value for $$\Omega$$, whilst ensuring the elements of the matrix are contrained as I described? Any advice would be very welcome.