LDL decomposition of a symmetric matrix

First step. $L_2\leftarrow L_2-(1/2)L_1$: multiplier $\ell_{21}=1/2$. New matrix:
$$\begin{pmatrix}4&2\\0&4\end{pmatrix}.$$

Decomposition.
$$L=\begin{pmatrix}1&0\\1/2&1\end{pmatrix},\qquad D=\begin{pmatrix}4&0\\0&4\end{pmatrix}.$$

Verify.
$$LDL^T=\begin{pmatrix}1&0\\1/2&1\end{pmatrix}\begin{pmatrix}4&0\\0&4\end{pmatrix}\begin{pmatrix}1&1/2\\0&1\end{pmatrix}=\begin{pmatrix}4&2\\2&5\end{pmatrix}.$$
Correct.

Conclusion. $A=LDL^T$ with the matrices above.