According to one source, Alexander is believed to have been killed by Indians.

 It is believed that while living in Ireland, the family name was Sarsfield, a landed gentry family of considerable note. The Sarsfield family in Ireland were "Militant" Catholics rather than Huguenots, and that is possible that an anti-catholic branch of the family "split off" and changed the name to Chesnut. Alexander Chesnut left Ireland and settled in Frederick Co. Virginia, which later became Hampshire Co., West Virginia. There is some documentary evidence of a deed or lease and related power of attorney recorded in Hampshire Co involving land originaly deeded to an Alexander Chesnut (called McChesnut) in which John of South Carolina is described as his son and heir.

Here is a letter from the Camden Archives in SC:
Letter from the Camden Archives Chesnut Family File --from Henry S. Chesnut in Louisville, Kentucky, dated September 23, 1965, to Mrs. Arthur Metts in Camden, South Carolina:
Dear Mrs. Metts:
I am happy to have your letter of September 18. I had written the County Court Clerk at Camden to inquire if he knew of anyone interested in the history of the Chesnut family of Camden.
We have done research in South Carolina, Virginia, and the Ulster Plantation of Northern Ireland.
The Chesnut family arrived in the Barony of Dunluce, County Andrim, Northern Ireland in 1690. That is as far back as we can presently establish and prove. Samuel Chesnut was the head of the family. The Chesnuts and the McKinneys jointly held "Carnaff" which was the name of the townland in Derrykeighan Parish there. Samuel Chesnut is buried at the south wall of the church. His gravestone reads 'killed while ploughing 1734'. The present sexton advises that some one brings flowers periodically to the grave but he does not know the name of the person.
We know of four sons of Samuel who came to America: Alexander, James, William, and David. There may have been more. Alexander (your ancestor) settled in Hampshire County, Virginia (now West Virginia) where he was killed by the Indians. David (my ancestor) settled in Chester County, South Carolina. William settled in Louisa County, Virginia, and a son John went to Kentucky. James settled somewhere in North Carolina.
David has three sons, all born in County Antrim: Samuel, the eldest, named for David's father, of course, and James and Alexander, named for David's brothers as above stated. David died in Chester County, S.C. in 1779 and is buried there. Samuel died in Richland County, S.C., James died in Chester County, S.C., and Alexander died in Todd County, Kentucky in 1809. Samuel, James and Alexander all served in the South Carolina Militia during the Revolution in Captain John Turner's company in Colonel Richard Winn's Regiment and their records of service are in the South Carolina Archives.
I plan to go to Northern Ireland next summer and expect to do intensive research on the family. Recently, I have had the Ulster-Scot Historical Society of Belfast do research and make a report and I would be happy to make photo copies of that report and send to you if you would care for it. Perhaps it would be helpful to your niece in Londonderry in her study of the family.
The Sarsfield name has been a legend in our branch of the family, but I have not been able to pin it down as to its origin. Dr. Wiley G. Chesnut, of Miami, Oklahoma, is building quite an estate on the lake near Miami and he is naming it "Sarsfield". Dr. Chesnut is from our branch of the family and he has no proof of the name. He thinks it would be a good idea to perpetuate the name.
The research on my branch of the Chesnut family in South Carolina was done by Mrs. Louise Kelly Crowder (Mrs. Dr. James W.) of Chester. It is most elaborate and supported by many authentic exhibits. It begins with David and would be no assistance to your line, although it may be interesting reading.
Those of the family who came to Virginia highlands and to South Carolina spell the name Chesnut. It is so spelled in the beginning in Northern Ireland and in the King James version of the Bible in the Book of Ezekiel. Within the last century in Northern Ireland and in this country, some have taken one or two extra t's. Miss Alice M. Chestnutt of Gracehill, Ballymena, County Antrim, Northern Ireland, (age 85) tells me that her grandfather added one t in the middle and that her father added another on the end.

Henry S. Chesnut