Lusztig data of Kashiwara-Nakashima tableaux in type D

We describe the embedding from the crystal of Kashiwara-Nakashima tableaux in type $D$ of an arbitrary shape into that of $\mathbf{i}$-Lusztig data associated to a family of reduced expressions $\mathbf{i}$ which are compatible with the maximal Levi subalgebra of type $A$. The embedding is described explicitly in terms of well-known combinatorics of type $A$ including the Sch\"{u}tzenberger's jeu de taquin and an analog of RSK algorithm.
Comment: 35 pages, typos corrected, comments added on Remark 5.3 and Remark 5.4, to appear in Algebras and Representation Theory