e embeded in
深深的印在,例如：That terrible day will be embeded in the children's memory.(喻)那可怕的一天将深深印在孩子们的记忆中.
埋藏,例如：Rocks of assorted sizes and shapes are embeded in the bottom of a glacier.多种尺寸和形状的岩石都埋藏在冰川的底部.
镶嵌,例如：On the wedding I saw blooms embeded in the bed of the bedroom.在婚礼上,我看见卧室的床上镶嵌着花朵.
嵌入,例如：It can be embeded into various communication systems flexibly.它可以嵌入到各种通信系统灵活.
be entrenched in
盘踞,例如：In 1719,Qing government troops were sent into Tibet to dispel the Zungar forces which had been entrenched in Lhasa for three years.1719年,清政府派军队进入西藏,驱逐盘踞拉萨三年之久的准噶尔部.
be ingrained in
根深蒂固,例如：The Depression of 1929-40 will be ingrained in the minds of Americans for generations to come.40的经济萧条将会根深蒂固的印在美国未来几代的思想里.
据我所知,用be embeded/entrenched/ingrained in (one's) mind 是都是“根深蒂固与某人的思想”的意思,其他情况直译/意译的话都不太合适.
