Reversible AJW-algebras

The main result states that every special AJW-algebra can be decomposed into the direct sum of totally irreversible and reversible subalgebras. In turn, every reversible special AJW-algebra decomposes into a direct sum of two subalgebras, one of which has purely real enveloping real von Neumann algebra, and the second one contains an ideal, whose complexification is a C$^*$-algebra and the annihilator of this complexification in the enveloping $C^*$-algebra of this subalgebra is equal to zero.