The Lie algebra su(3) direct sum su(3)

The real Lie algebra $\mathfrak{su}(3)\oplus\mathfrak{su}(3)$ is the direct sum of two copies of $\mathfrak{su}(3)$ , with

[(X_1,X_2),(Y_1,Y_2)]=([X_1,Y_1],[X_2,Y_2]).

It has real dimension $16$ . The two eight-dimensional factors are commuting simple ideals, so the sum is semisimple and not simple.

Magic-square entry

For the compact real convention,

\mathfrak M(\mathbb C,\mathbb C)\cong \mathfrak{su}(3)\oplus\mathfrak{su}(3).

It is the nonsimple entry of the square. Each complex input is being used as a real algebra.

References

John C. Baez, “The Octonions,” §4.3, Table 5, complex–complex entry. Checked section.

Input	$\mathbb R$	$\mathbb C$	$\mathbb H$	$\mathbb O$
$\mathbb R$	$\mathfrak{so}(3)$	$\mathfrak{su}(3)$	$\mathfrak{sp}(3)$	$\mathfrak f_4$
$\mathbb C$	$\mathfrak{su}(3)$	$\mathfrak{su}(3)\oplus\mathfrak{su}(3)$	$\mathfrak{su}(6)$	$\mathfrak e_6$
$\mathbb H$	$\mathfrak{sp}(3)$	$\mathfrak{su}(6)$	$\mathfrak{so}(12)$	$\mathfrak e_7$
$\mathbb O$	$\mathfrak f_4$	$\mathfrak e_6$	$\mathfrak e_7$	$\mathfrak e_8$

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