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There is an exercise on page 114 of Humphreys' Linear Algebraic Groups (GTM 21)

Prove that $SO(3, \mathbb R)$ (= group of $3 \times 3$ real orthogonal matrices of determinate $1$) is a connected subgroup of $SL(3, \mathbb C)$ consisting of semisimple elements, but not commutative.

The semisimplicity of the elements and the noncommutativity of the group is clear. For the connectedness, I think this proposition might be applied:

Let $G$ be an algebraic group, $I$ an idex set, $f_i: X_i \rightarrow G$ $(i \in I)$ a family of morphisms from irreducible varieties $X_i$, such that $e \in Y_i = f_i(X_i)$ for each $i \in I$. Set $M = \cup_{i \in I} Y_i$. Then the intersection of all closed subgroup of $G$ containing $M$ is a connected subgroup of $G$.

But I don't know how to determine the $X_i$'s.

Thanks for help.

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    @Andrea Mori: Thank you very much for the geometric answer~ Thanks a lot for taking time to explain it to me.2012-02-13

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