Definition: The Magnus’ bracket is defined as 
Properties:
- for a power series
one has
- Hausdorf:
,
,
where the Hausdorff polarization operator
is defined as following:
.
- Lemma: For two power series
and
s/t
one has the following equivalence:
.
.
For 
where
where
,
,
.
![z = x + y + \frac12 [x,y] + \frac1{12} \{x,y^2\}_M + \frac1{12} \{y,x^2\}_M -\frac1{24} [\{x,y^2\}_M,x] - \frac1{720} \{x,y^4\}_M - \frac1{720} \{y,x^4\}_M + \frac1{180} [[\Delta_1,x],y] - \frac1{180}\{\Delta_1,y^2\}_M - \frac1{120} [\Delta_1,\Delta] - \frac1{360} [\Delta_2,\Delta] + \cdots](https://s0.wp.com/latex.php?latex=z+%3D+x+%2B+y+%2B+%5Cfrac12+%5Bx%2Cy%5D+%2B+%5Cfrac1%7B12%7D+%5C%7Bx%2Cy%5E2%5C%7D_M+%2B+%5Cfrac1%7B12%7D+%5C%7By%2Cx%5E2%5C%7D_M+-%5Cfrac1%7B24%7D+%5B%5C%7Bx%2Cy%5E2%5C%7D_M%2Cx%5D+-+%5Cfrac1%7B720%7D+%5C%7Bx%2Cy%5E4%5C%7D_M+-+%5Cfrac1%7B720%7D+%5C%7By%2Cx%5E4%5C%7D_M+%2B+%5Cfrac1%7B180%7D+%5B%5B%5CDelta_1%2Cx%5D%2Cy%5D+-+%5Cfrac1%7B180%7D%5C%7B%5CDelta_1%2Cy%5E2%5C%7D_M+-+%5Cfrac1%7B120%7D+%5B%5CDelta_1%2C%5CDelta%5D+-+%5Cfrac1%7B360%7D+%5B%5CDelta_2%2C%5CDelta%5D+%2B+%5Ccdots&bg=ffffff&fg=000000&s=0 "z = x + y + \frac12 [x,y] + \frac1{12} \{x,y^2\}_M + \frac1{12} \{y,x^2\}_M -\frac1{24} [\{x,y^2\}_M,x] - \frac1{720} \{x,y^4\}_M - \frac1{720} \{y,x^4\}_M + \frac1{180} [[\Delta_1,x],y] - \frac1{180}\{\Delta_1,y^2\}_M - \frac1{120} [\Delta_1,\Delta] - \frac1{360} [\Delta_2,\Delta] + \cdots")
The Magnus’ formula for ordered exponentials
And this is really beautiful thing: for the ordered exponential 
where 
![\frac{d\Omega(t)}{dt} = \{A, \frac{\Omega}{1-e^{-\Omega}}\}_M = \sum_{n=0}^\infty \beta_n \{A,\Omega^n\}_M = A + \frac12 [A,\Omega] + \frac1{12}\{A,\Omega^2\}_M + \cdots](https://s0.wp.com/latex.php?latex=%5Cfrac%7Bd%5COmega%28t%29%7D%7Bdt%7D+%3D+%5C%7BA%2C+%5Cfrac%7B%5COmega%7D%7B1-e%5E%7B-%5COmega%7D%7D%5C%7D_M+%3D+%5Csum_%7Bn%3D0%7D%5E%5Cinfty+%5Cbeta_n+%5C%7BA%2C%5COmega%5En%5C%7D_M+%3D+A+%2B+%5Cfrac12+%5BA%2C%5COmega%5D+%2B+%5Cfrac1%7B12%7D%5C%7BA%2C%5COmega%5E2%5C%7D_M+%2B+%5Ccdots&bg=ffffff&fg=000000&s=0 "\frac{d\Omega(t)}{dt} = \{A, \frac{\Omega}{1-e^{-\Omega}}\}_M = \sum_{n=0}^\infty \beta_n \{A,\Omega^n\}_M = A + \frac12 [A,\Omega] + \frac1{12}\{A,\Omega^2\}_M + \cdots")
Source: T1264 = W. Magnus “On the exponential solution of differential equations for a linear operator”; Communications of Pure and Applied mathematics, Vol VII, 6490673 (1954). Note: This paper is probably mother of the whole this area of matrix exponentials.
Related here: Magnus’ Bracket Operator – https://eikonal.wordpress.com/2010/02/15/magnus-bracket-operator/
