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i have a consideration to post. Let V be a vector space over Reals with a symmetric inner product < , > : V x V -> R. We can introduce the concept of formal transposition of linear operator X on V, [tex]\forall x,y \in V[/tex] by the equation

[tex]

\left\langle x , X y \right\rangle = \left\langle X^t x , y \right\rangle

[/tex]

Now recalling that the orthogonal group and the special orthogonal group are given by

[tex]

O(V) = \left\{ X | X^t = X^{-1} \right\} = \left\{ X | det X = \pm 1 \right\}

[/tex]

[tex]

SO(V) = \left\{ X | X^t = X^{-1}, det X = 1 \right\}

[/tex]

It can be proven that these groups are closed Lie Groups, in the convergence sense with the distance induced by usual operators norm, and further that the associated Lie Algebras, namely o(V) and so(V), are the same.

How this can be possible? O(V) should contain SO(V) as a subgroup and the difference between them are "rotations" which do not preserve orientation (i.e. parity). And more, if this is true how can i build up an exponential map from an element of the algebra to recover a "rotation" belonging to O(V) but not to SO(V)?

The technical aspects are clear to me, im looking for some clarification from the concrete point of view...thank you.