Editing Rotation matrix (section)

=== Baker–Campbell–Hausdorff formula ===
{{main|Baker–Campbell–Hausdorff formula|Rotation group SO(3)#Baker–Campbell–Hausdorff formula}}
The BCH formula provides an explicit expression for {{math|1=''Z'' = log(''e''<sup>''X''</sup>''e''<sup>''Y''</sup>)}} in terms of a series expansion of nested commutators of {{mvar|X}} and {{mvar|Y}}.<ref>{{Harvnb|Hall|2004|loc=Ch.&nbsp;3}}; {{Harvnb|Varadarajan|1984|loc=§2.15}}</ref> This general expansion unfolds as<ref group=nb>For a detailed derivation, see [[Derivative of the exponential map]]. Issues of convergence of this series to the right element of the Lie algebra are here swept under the carpet. Convergence is guaranteed when {{math|{{norm|''X''}} + {{norm|''Y''}} < log 2}} and {{math|{{norm|''Z''}} < log 2}}. If these conditions are not fulfilled, the series may still converge. A solution always exists since {{math|exp}} is onto{{clarify|date=June 2017}} in the cases under consideration.</ref>
:<math> Z = C(X, Y) = X + Y + \tfrac{1}{2} [X, Y] + \tfrac{1}{12} \bigl[X,[X,Y]\bigr] - \tfrac{1}{12} \bigl[Y,[X,Y]\bigr] + \cdots .</math>

In the {{nowrap|3 × 3}} case, the general infinite expansion has a compact form,<ref>{{Harv|Engø|2001}}</ref>
:<math>Z = \alpha X + \beta Y + \gamma[X, Y],</math>
for suitable trigonometric function coefficients, detailed in the [[Rotation group SO(3)#Baker–Campbell–Hausdorff formula|Baker–Campbell–Hausdorff formula for SO(3)]].

As a group identity, the above holds for ''all faithful representations'', including the doublet (spinor representation), which is simpler. The same explicit formula thus follows straightforwardly through Pauli matrices; see the [[Pauli matrices#Exponential of a Pauli vector|{{nowrap|2 × 2}} derivation for SU(2)]]. For the general {{math|''n'' × ''n''}} case, one might use Ref.<ref>{{Cite journal | doi = 10.3842/SIGMA.2014.084|last1 = Curtright|last2 = Fairlie|last3 = Zachos |first1 = T L |first2 = D B |first3 = C K|author-link=Thomas Curtright|author-link2=David Fairlie|author-link3=Cosmas Zachos|year = 2014|title = A compact formula for rotations as spin matrix polynomials| journal =SIGMA| volume=10| page=084|arxiv = 1402.3541|bibcode = 2014SIGMA..10..084C|s2cid = 18776942}}</ref>