The Balfour Declaration. The "National Home for the Jews" encompassed both areas under the British Mandate until 2/3 was severed and given to the Hashemite clan in 1922:
The Balfour Declaration — Commentary
On November 2, 1917, Britain issued the Balfour Declaration:
His Majesty's Government views with favor the establishment in Palestine of a national home for the Jewish people, and will use their best endeavors to facilitate the achievement of this object, it being clearly understood that nothing shall be done which may prejudice the civil and religious rights of existing non-Jewish communities in Palestine or the rights and political status enjoyed by Jews in any other country.
According to the Peel Commission, appointed by the British Government to investigate the cause of the 1936 Arab riots, "the field in which the Jewish National Home was to be established was understood, at the time of the Balfour Declaration, to be the whole of historic Palestine, including Transjordan."
The Mandate for Palestine's purpose was to put into effect the Balfour Declaration. It specifically referred to "the historical connections of the Jewish people with Palestine" and to the moral validity of "reconstituting their National Home in that country." The term "reconstituting" shows recognition of the fact that Palestine had been the Jews' home. Furthermore, the British were instructed to "use their best endeavors to facilitate" Jewish immigration, to encourage settlement on the land and to "secure" the Jewish National Home. The word "Arab" does not appear in the Mandatory award.
The Mandate was formalized by the 52 governments at the League of Nations on July 24, 1922.
But for the record, I never said that they promised the Jews a state on the entire area west of the Jordan river. I said that they had written to Arab leaders that they had not promised them a state west of the Jordan river. These are not the same thing.
But since you brought it up, I checked, and I was slightly inaccurate. They had actually explicitly denied that they had promised the Arabs independence at all in Palestine.
Here is what Sir Henry MacMahon, High Commissioner for Egypt, wrote to the Sharif of Mecca in 1915: "I feel it my duty to state, and I do so definitely and emphatically, that it was not intended by me in giving this pledge to King Hussein to include Palestine in the area in which Arab independence was promised. I also had every reason to believe at the time that the fact that Palestine was not included in my pledge was well understood by King Hussein."