# The BCH formula and the symmetric BCH formula up to terms of degree 20

## by Fernando Casas and Ander Murua

## The BCH formula written in a basis of the free Lie algebra

log(exp(X) exp(Y)) = X + Y + 1/2 [X,Y] + 1/12 [X,[X,Y]] - 1/12 [Y,[X,Y]] - 1/24[Y,[X,[X,Y]]] + ...

## The symmetric BCH formula written in a basis of the free Lie algebra

log(exp(X/2) exp(Y) exp(X/2)) = X + Y - 1/24 [X,[X,Y]] - 1/12 [Y,[X,Y]] + ...

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