The Bunch Kaufman decomposition of a hermitian matrix $A$ is such that:
$$P A P^\top = LBL^H,$$
where $B$ is a block diagonal matrix, with $1\times 1$ or $2 \times 2 $ diagonal blocks, and
$L$ is a unit lower triangular matrix.
The Cholesky decomposition with diagonal $D$ of a hermitian matrix $A$ is such that:
$$A = LDL^H,$$
where $D$ is a diagonal matrix, and $L$ is a unit lower triangular matrix.
The Cholesky decomposition of a hermitian positive definite matrix $A$ is such that:
$$A = LL^H,$$
where $L$ is a lower triangular matrix.