  Let $\Omega\subset \mathbb{R}^4$ be a smooth bounded domain,
$W_0^{2,2}(\Omega)$ be the usual Sobolev space. For any positive integer
$\ell$, $\lambda_{\ell}(\Omega)$ is the $\ell$-th eigenvalue of the
bi-Laplacian operator. Define $E_{\ell}=E_{\lambda_1(\Omega)}\oplus
E_{\lambda_2(\Omega)}\oplus\cdots\oplus E_{\lambda_{\ell}(\Omega)}$, where
$E_{\lambda_i(\Omega)}$ is eigenfunction space associated with
$\lambda_i(\Omega)$. $E^{\bot}_{\ell}$ denotes the orthogonal complement of
$E_\ell$ in $W_0^{2,2}(\Omega)$. For $0\leq\alpha<\lambda_{\ell+1}(\Omega)$, we
define a norm by $\|u\|_{2,\alpha}^{2}=\|\Delta u\|^2_2-\alpha \|u\|^2_2$ for
$u\in E^\bot_{\ell}$. In this paper, using the blow-up analysis, we prove the
following Adams inequalities $$\sup_{u\in E_{\ell}^{\bot},\,\|
u\|_{2,\alpha}\leq 1}\int_{\Omega}e^{32\pi^2u^2}dx<+\infty;$$ moreover, the
above supremum can be attained by a function $u_0\in E_{\ell}^{\bot}\cap
C^4(\overline{\Omega})$ with $\|u_0\|_{2,\alpha}=1$.
  This result extends that of Yang (J. Differential Equations, 2015), and
complements that of Lu and Yang (Adv. Math. 2009) and Nguyen (arXiv:
1701.08249, 2017).
