Rotation of coordinate system

[cupuyc]

Hello. Sorry for my english.

I have 3 coordinate systems [math]OI[/math], [math]OK[/math], [math]OE[/math]. System [math]OK[/math] defined  in [math]OI[/math] by quaternion [math]A[/math].  System [math]OE[/math] defined in [math]OI[/math] by quaternion [math]B[/math]. I need to find quaternion [math]C[/math] that define rotation from [math]OE[/math] to [math]OK[/math].

My solution. 2 rotations [math]OI \to OK[/math] and [math]OI \to OE \to OK[/math] are equal.

[math]A = B \circ C[/math],

and

[math]\tilde{B} \circ A = \tilde{B} \circ B \circ C = C[/math],

i.e.

[math]C = \tilde{B} \circ A[/math].

Is it right?