I am trying to write the D'Alembert operator after 3+1 splitting. We have,

\[\Box=g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}\]

I can decompose as

\[\Box=(h^{ab}e^{\alpha}_{a}e^{\beta}_{b}-n^{\alpha}n^{\beta})\nabla_\alpha\nabla_\beta\]

where the Greek indices are 4-dim and latin indices are 3-dim. After a bit of algebra I get,

\(\Box=h^{ab}D_a D_b-h^{ab}(D_a e^\beta_b)\nabla_\beta-n^\alpha n^\beta\nabla_\alpha\nabla_\beta\)

where D is covariant derivative in 3-dim. I wonder if the last two terms are vanishing somehow. Or generally what's the form of D'Alembert operator in ADM formalism.