Step 1: Recall the trend of metallic character.
Metallic character increases:
(i) from right to left across a period
and
(ii) from top to bottom in a group
Elements with greater tendency to lose electrons show higher metallic character.
Step 2: Analyze the given elements.
\[
B
\]
(Boron) is a metalloid and has the least metallic character among the given elements.
\[
Al
\]
(Aluminium) is a metal and is more metallic than boron.
\[
Mg
\]
(Magnesium) lies further left in the same period compared to aluminium, so it is more metallic than aluminium.
\[
K
\]
(Potassium) is an alkali metal and has the highest metallic character because it easily loses one electron.
Step 3: Arrange in increasing order.
Thus, the increasing order of metallic character is
\[
B\lt Al\lt Mg\lt K
\]
Step 4: Final conclusion.
Therefore, the correct arrangement is
\[
\boxed{B\lt Al\lt Mg\lt K}
\]