Well a Lutheran is one who (as it notes in the congregational model constitution):
*C2.05. This congregation accepts the Unaltered Augsburg Confession as a true witness to the Gospel [emphasis mine], acknowledging as one with it in faith and doctrine all churches that likewise accept the teachings of the Unaltered Augsburg Confession.
*C2.06. This congregation accepts the other confessional writings in the Book of Concord, namely, the Apology of the Augsburg Confession, the Smalcald Articles and the Treatise, the Small Catechism, the Large Catechism, and the Formula of Concord, as further valid interpretations of the faith of the Church.
In other words, we hold the Gospel in one hand and the Book of Concord (BoC) in the other and these documents together form the basis of all our theological truths.
This Book of Concord is a book that every Lutheran should have and have read cover to cover. In fact, I'll go further and note that every Christian should have and read it as it explains much about what it is to be a Christian that every Christian should know.
At this writing, because of the acceptance of the CCM and the various doctrinal problems it poses, neither the Missouri Synod or the Wisconsin Synod consider the ELCA to be a Lutheran denomination.