I look into the original STAN code of “cor_lagsar” in BRMS and have a question about formulating lpdf.

real normal_lagsar_lpdf(vector y, vector mu, real sigma,
real rho, data matrix W, data vector eigenW) {
int N = rows(y);
real inv_sigma2 = inv_square(sigma);
matrix[N, N] W_tilde = add_diag(-rho * W, 1);
vector[N] half_pred;
real log_det;
half_pred = W_tilde * y - mu;
log_det = sum(log1m(rho * eigenW));
return 0.5 * N * log(inv_sigma2) + log_det -
0.5 * dot_self(half_pred) * inv_sigma2;
}

All are clear, except for computing log density by " log_det = sum(log1m(rho * eigenW))“.
Why use “eigenW”, rather than the original W matrix, here?
I notice that the bounded spatial lag coefficient is defined in the data transformation part.